# Graph Neural Networks for reconstruction and classification in KM3NeT

S.Reck,11footnotetext: Corresponding author. D. Guderian  G. Vermariën  A.
Domi

###### Abstract

KM3NeT, a neutrino telescope currently under construction in the Mediterranean
Sea, consists of a network of large-volume Cherenkov detectors. Its two
different sites, ORCA and ARCA, are optimised for few GeV and TeV-PeV neutrino
energies, respectively. This allows for studying a wide range of physics
topics spanning from the determination of the neutrino mass hierarchy to the
detection of neutrinos from astrophysical sources.

Deep Learning techniques provide promising methods to analyse the signatures
induced by charged particles traversing the detector. This document will cover
a Deep Learning based approach using Graph Convolutional Networks to classify
and reconstruct events in both the ORCA and ARCA detector. Performance studies
on simulations as well as applications to real data will be presented,
together with comparisons to classical approaches.

##  1 Introduction

KM3NeT is a network of neutrino telescopes under construction in the
Mediterranean Sea [1], consisting of two detectors: ORCA, optimised for
oscillation studies with atmospheric neutrinos in the 1 to 100 GeV range, and
ARCA, optimised for cosmic neutrino searches in the TeV to PeV energy range.
Charged particles produced in neutrino interactions with water emit Cherenkov
light, which is detected by 3" photomultiplier tubes (PMTs) hosted in high-
pressure glass spheres (DOMs). 31 PMTs are stored in a single DOM, and 18 DOMs
are fixed to a long vertical detection unit (DU) that is anchored to the sea
floor. Each of the three building blocks of KM3NeT will consist out of 115
DUs. The detected light on each PMT (hit) can be used for reconstructing the
particles’ properties, such as their energy or direction. A hit holds
information about the time and the xyz-position of the PMT that recorded the
light, as well as its pointing direction. Together, the hits of an event are
the input to reconstruction and classification algorithms. Since the data
recorded by KM3NeT closely resembles point clouds, Graph Neural networks
(GNNs) are a natural choice for the architecture.

In the input to the GNN, the information of each single hit becomes the node
feature [2]. The architecture of the GNN used in this work resembles the
ParticleNet model proposed by Qu et al. [3]. It consists out of three Edge
Convolutional blocks [4], followed by a global pooling layer and two fully
connected layers. A custom open-source implementation for this architecture
was developed using the tensorflow [5] back end222See
https://github.com/StefReck/MEdgeConv..

##  2 Atmospheric muon reconstruction

KM3NeT’s primary goal is to study neutrinos. However, the majority of detected
events are atmospheric muons produced in cosmic ray induced air showers, which
usually pose as a background for neutrino measurements. But they provide a
wide range of applications as well, like measuring the cosmic ray composition.
GNNs can be used to reconstruct the zenith angle and the muon multiplicity of
an incident muon or muon bundle. For this, a set of 25 million atmospheric
muon events was simulated using the software package MUPAGE [6] for the
initial four-line set-up of the ORCA detector. Figure 1 shows the zenith angle
reconstruction performance in comparison to a classical maximum-likelihood
based algorithm for single tracks as described in [1]. Since it was trained on
the expected distribution, the deep learning reconstruction is biased for true
cosine zenith below 0.5, leading to an increase in the error there. GNNs
provide a comparable precision to classical methods on atmospheric muon events
and a substantial reduction of the median zenith error by a factor of 2 for
the rarer multi-muon events, which make up about a fifth of the dataset. Since
the classical method was not optimised for multi-muon events, its performance
could potentially be improved, e.g. by an adjustment of the used PDF.

(a) all events

(b) events with two or more muons

Figure 1:  Absolute difference between reconstructed and true zenith angle
plotted over the true zenith angle for selected atmospheric muons in ORCA4.
Shown are the median and the 68% band for the classical reco (orange) and the
deep learning reco (blue).

The number of atmospheric muons in an event (muon multiplicity) can be used
for the identification of primary particles. This deep learning approach is
the first reconstruction of the muon multiplicity in KM3NeT. Even though the
detector is still in an early stage of construction, deep learning can already
provide a good estimator for the muon multiplicity (see Figure 2). Figure 3
shows the result of applying the trained network on 250,000 atmospheric muon
events simulated with Corsika [7] SIBYLL 2.3c [8] using the GST-3 [9]
spectrum. As can be seen, the current detector already provides a decent
separation power between iron and proton induced events.

Figure 2:  Reconstructed versus true muon multiplicity for events with a
similar number of hits (500 to 600). Even without use of the correlation
between the multiplicity and the number of hits, the reconstruction is
meaningful.

Figure 3:  Reconstructed (light) and true (dark) muon multiplicity rates for
events generated in proton (blue) and iron (orange) induced atmospheric
showers.

##  3 Studies on neutrino selections

GNNs can be used to select neutrino events in ORCA4 real data. To this end,
different networks have been trained on simulated neutrino data [10] to
perform classification and reconstruction tasks. The cuts are optimized on MC
data for the highest possible neutrino yield while keeping a low background
contamination. The application to real data shows a reasonable agreement
between expectation and observation, proving to be a valuable method to cross-
check conventional methods.

The ParticleNet used this far is implemented in Tensorflow. An alternative
architecture using PyTorch Geometric [11] has also been tested. The benefit of
this framework is that it has many implementations of GNN layers included,
which allow for easy prototyping of different architectures. As a first
benchmark for the framework, ParticleNet is implemented and it is trained for
the signal/background classifier on ORCA4. The accuracy of both
implementations can be seen in Figure 4. The difference between the
implementations can be attributed to the use of more advanced learning rate
schedulers in Tensorflow. The shape of the accuracy curve is identical between
the two implementations.

Figure 4: A comparison of the accuracy for background muon classification
between PyTorch and Tensorflow for the same ParticleNet architecture [3].

Figure 5: Normed distribution of the neutrino_score for true muon (blue-
dashed) and true neutrino (red-continuous) events for the full ARCA detector.

GNNs can also be used for selecting cosmic neutrinos in ARCA over the muon
background. This can be achieved by cutting over a GNN neutrino_score,
representing the probability of an event being induced by a neutrino. The
selected neutrinos are then further classified into tracks and showers by a
subsequent GNN classification step. Figure 5 shows the distribution of the
neutrino_score assigned by the GNN to true neutrinos and true atmospheric
muons, for the full ARCA detector. The GNN performances are now being compared
with other ML methods with the goal of finding the most promising method for
cosmic neutrino selection.

## References

  * [1] S. Adrián-Martínez et al. (KM3NeT Collaboration), _Letter of intent for KM3NeT 2.0_ , _Journal of Physics G: Nuclear and Particle Physics, 43 (8), 084001, 2016_ [1601.07459v2].
  * [2] J. Zhou, G. Cui, Z. Zhang, C. Yang, Z. Liu and M. Sun, _Graph neural networks: A review of methods and applications_ , 1812.08434.
  * [3] H. Qu and L. Gouskos, _ParticleNet: Jet tagging via particle clouds_ , _Phys. Rev. D 101, 056019 (2020)_ [1902.08570v3].
  * [4] Y. Wang, Y. Sun, Z. Liu, S.E. Sarma, M.M. Bronstein and J.M. Solomon, _Dynamic graph CNN for learning on point clouds_ , 1801.07829.
  * [5] M. Abadi et al., _TensorFlow: Large-scale machine learning on heterogeneous distributed systems_ , 1603.04467v2.
  * [6] G. Carminati, A. Margiotta and M. Spurio, _Atmospheric muons from parametric formulas: a fast generator for neutrino telescopes (MUPAGE)_ , _Comput.Phys.Commun.179:915-923,2008_ [0802.0562v2].
  * [7] D. Heck, J. Knapp, J. Capdevielle, G. Schatz and T. Thouw, _CORSIKA: A Monte Carlo code to simulate extensive air showers_ , _FZKA-6019_ (1998) .
  * [8] F. Riehn, H.P. Dembinski, R. Engel, A. Fedynitch, T.K. Gaisser and T. Stanev, _The hadronic interaction model SIBYLL 2.3c and feynman scaling_ , _PoS ICRC2017 (2017) 301_ [1709.07227v1].
  * [9] T.K. Gaisser, T. Stanev and S. Tilav, _Cosmic ray energy spectrum from measurements of air showers_ , _Frontiers of Physics_ (2013)  [1303.3565v1].
  * [10] S. Aiello, A. Albert, S.A. Garre, Z. Aly, F. Ameli, M. Andre et al., _gseagen: The km3net genie-based code for neutrino telescopes_ , _Computer Physics Communications_ 256 (2020) 107477.
  * [11] M. Fey and J.E. Lenssen, _Fast graph representation learning with PyTorch Geometric_ , in _ICLR Workshop on Representation Learning on Graphs and Manifolds_ , 2019.


# Cell decision-making through the lens of Bayesian learning

Arnab Barua  Departement de Biochimie, Université de Montréal, Québec, Canada
Centre Robert-Cedergren en Bio-informatique et Génomique, Université de
Montréal, Québec, Canada  Haralampos Hatzikirou  Technische Univesität
Dresden, Center for Information Services and High Performance Computing,
Nöthnitzer Straße 46, 01062, Dresden, Germany  Mathematics Department, Khalifa
University, P.O. Box: 127788, Abu Dhabi, UAE

###### Abstract

Cell decision-making refers to the process by which cells gather information
from their local microenvironment and regulate their internal states to create
appropriate responses. Microenvironmental cell sensing plays a key role in
this process. Our hypothesis is that cell decision-making regulation is
dictated by Bayesian learning. In this article, we explore the implications of
this hypothesis for internal state temporal evolution. By using a timescale
separation between internal and external variables on the mesoscopic scale, we
derive a hierarchical Fokker-Planck equation for cell-microenvironment
dynamics. By combining this with the Bayesian learning hypothesis, we find
that changes in microenvironmental entropy dominate cell state probability
distribution. Finally, we use these ideas to understand how cell sensing
impacts cell decision-making. Notably, our formalism allows us to understand
cell state dynamics even without exact biochemical information about cell
sensing processes by considering a few key parameters.

###### keywords:

Cell decision-making; Bayesian learning; Least microEnvironmental Uncertainty
Principle (LEUP); Hierarchical Fokker-Planck equation; Cell sensing dynamics

##  1 Introduction

Decision-making is the process of choosing different actions based on certain
goals [1]. Similarly, cells make decisions as a response to microenvironmental
signals [2]. When external cues, such as signalling molecules, are received by
the cell where a series of chemical reactions is triggered inside the cell
[3]. This decision-making process is influenced by intrinsic signal
transduction pathways [4], the genetic cell network [5], extrinsic cues [6],
and molecular noise [7]. In turn, such intracellular regulation produces an
appropriately diverse range of decisions, in the context of differentiation,
phenotypic plasticity, proliferation, migration, and apoptosis. Understanding
the underlying principles of cellular decision-making is essential to
comprehend the behaviour of complex biological systems.

Cell sensing is a fundamental process that enables cells to respond to their
environment and make decisions. Typically, receptors on the cell membrane can
detect various stimuli such as changes in temperature [8], pH [9] or the
presence of specific molecules. The specificity of the receptors and the
signalling pathways that are activated are critical in determining the
response of the cell. However, receptors are not the sole sensing unit of the
cell. Recent studies have also revealed that cells use mechanical cues to make
decisions about their behaviour [10]. For example, cells can sense the
stiffness of the substrate and they are growing on [11]. In turn, cells make
decisions about changing their shape, migration, proliferation or gene
expression, in the context of a phenomenon called mechanotransduction [12].
Errors in cell sensing can lead to possible pathologies such as cancer [13],
autoimmunity [14], diabetes [15] etc.

Bayesian inference or updating has been the main toolbox for general-purpose
decision-making[16]. In the context of cell decision-making, this mathematical
framework assumes that cells integrate new information and update their
internal state based on the likelihood of different outcomes [17]. Although
static Bayesian inference was the main tool for understanding cell decisions,
recently Bayesian forecasting has been additionally employed to understand the
dynamics of decisions [18]. In particular, in [19] Mayer et al. have used
dynamic Bayesian prediction to model the estimation of the future pathogen
distribution by adaptive immune cells. A dynamic Bayesian prediction model has
been also used for bacterial chemotaxis [20]. Finally, the authors have
developed the Least microEnvironmental Uncertainty Principle (LEUP) that
employs Bayesian-based dynamic theory for cell decision-making [21, 22, 23,
24].

To understand the stochastic dynamics of the cell-microenvironment system, we
focus on the mesoscopic scale and we derive a Fokker-Planck equation. Fokker-
Planck formalism has been developed to study the time-dependent probability
distribution function for the infamous Brownian motion under the influence of
a drift force[25]. Though, we can see nowadays a huge number of applications
of Fokker-Planck equations (linear and non-linear) across the disciplines [26,
27]. Here, we will additionally assume a time-scale separation between
internal and external variables [28]. Timescale separation has been studied
rigorously[29] from the microscopic point of view using Langevin equations. In
the case of cell decision-making, microscopic dynamics have been studied,
specifically in the context of active Brownian motion and cell migration using
Langevin equations[30, 31, 22]. Understanding dynamics induced by a timescale
separation at the mesoscopic scale, using Fokker-Planck equations, has been
studied only recently by S. Abe [32].

Specifically, we will assume a timescale separation where cell decision time,
when internal states evolve, is slower than the characteristic time of the
variables that belong to the cellular microenvironment. This assumption is
particularly valid for cell decision-making at the timescale of a cell cycle,
such as differentiation. The underlying molecular regulation underlying these
decisions may evolve over even for many cell cycles [33, 34]. When these
molecular expressions cross a threshold, then cell decision emerges.

The structure of our paper is as follows: In Sec. 2 we present the Bayesian
learning dynamics for cell decision-making. In turn, we derive a fluctuation-
dissipation relation and the corresponding continuous-time dynamics of
cellular internal states. After that in Sec. 3, we elaborate on the concept of
the Hierarchical Fokker-Planck equation in relation to cellular decision-
making and the underlying Bayesian learning process. In Sec. 4 we demonstrate
the use of a simple example of coarse-grained dynamics for cell sensing to
analyze the steady-state distribution of cellular states in two scenarios: (i)
in absence of cell sensing and (ii) when the cell sensor is ON. Then in Sec. 5
we connect this idea with the Least microenvironmental Uncertainty
Principle(LEUP) as a special case of Bayesian learning. Finally, in Sec. 6 we
conclude and discuss our results and findings.

##  2 Cell decision making as Bayesian learning

Cells decisions, here interpreted as changes in the cellular internal states
$\mathbf{X}$ within a decision time $\tau$, are realized via (i) sensing their
microenvironment $\mathbf{Y}$ and combining this information with (ii) an
existing predisposition about their internal state. In a Bayesian language,
the former can be interpreted as the empirical likelihood
$P\left(\mathbf{Y}\mid\mathbf{X}\right)$ and the latter as the prior
distribution $P\left(\mathbf{X}\right)$. Interestingly, we assume that the
cell tries to build increasingly informative priors over time that minimize
the cost of energy associated with sampling the cellular microenvironment. For
instance assuming that cell fate decisions follow such Baysian learning
dynamics, during tissue differentiation, we observe the microenvironment
evolving into a more organized state (e.g. pattern formation). Therefore, one
can observe a reduction of the microenvironmental entropy over time, which is
further associated with the microenvironmental probability distribution or
likelihood in Bayesian inference. Here we will postulate the cells evolve the
distribution of their internal states in the form of Bayesian learning.

###  2.1 A fluctuation-dissipation relation

Formalizing the above, let us assume that after a decision time, $\tau$ the
cell updates its state from $\mathbf{X}$ to $\mathbf{X^{\prime}}$. According
to Bayesian learning, the posterior of the previous time
$P\left(\mathbf{X}\mid\mathbf{Y}\right)$ becomes prior to the next time-step,
i.e. $P(\mathbf{X^{\prime}})=P\left(\mathbf{X}\mid\mathbf{Y}\right)$.
Therefore, the Bayesian learning dynamics read:

$\begin{split}&P\left(\mathbf{X^{\prime}}\right)=\frac{P\left(\mathbf{Y}\mid\mathbf{X}\right)P\left(\mathbf{X}\right)}{P\left(\mathbf{Y}\right)},\\\
&\implies\ln\frac{P\left(\mathbf{X^{{}^{\prime}}}\right)}{P\left(\mathbf{X}\right)}=\ln\frac{P\left(\mathbf{Y}\mid\mathbf{X}\right)}{P\left(\mathbf{Y}\right)}.\\\
&\implies\int
P\left(\mathbf{X^{{}^{\prime}}},\mathbf{X},\mathbf{Y}\right)\ln\left(\frac{P\left(\mathbf{X^{{}^{\prime}}}\right)}{P\left(\mathbf{X}\right)}\right)d\mathbf{X^{{}^{\prime}}}d\mathbf{X}d\mathbf{Y}\\\
&=\int
P\left(\mathbf{X^{{}^{\prime}}},\mathbf{X},\mathbf{Y}\right)\ln\left(\frac{P\left(\mathbf{Y}\mid\mathbf{X}\right)}{P\left(\mathbf{Y}\right)}\right)d\mathbf{X^{{}^{\prime}}}d\mathbf{X}d\mathbf{Y}\\\
&\implies
D\left(\mathbf{X^{{}^{\prime}}}\mid\mid\mathbf{X}\right)=\tilde{\beta}I\left(\mathbf{Y},\mathbf{X}\right),\end{split}$
(1)

where $\tilde{\beta}=\frac{\int
P\left(\mathbf{X^{{}^{\prime}}}\mid\mathbf{X},\mathbf{Y}\right)d\mathbf{X^{{}^{\prime}}}}{\int
P\left(\mathbf{X},\mathbf{Y}\mid\mathbf{X^{{}^{\prime}}}\right)d\mathbf{X}d\mathbf{Y}}$
which is different than one if the corresponding conditional distributions are
non-normalizable, such as power-laws or multimodal distributions. In the above
relation, the Kullback-Leibler divergence
$D\left(\mathbf{X^{{}^{\prime}}}\mid\mid\mathbf{X}\right)$ , that quantifies
the convergence to the equilibrium distribution of the internal value of
$\mathbf{X}$, is connected to the amount of available information
$I\left(\mathbf{Y},\mathbf{X}\right)$ between the cell and its
microenvironment. From Eq.(1), the Kullbeck-Leibler divergence can be further
elaborated in terms of Fisher information as

$\begin{split}&D\left(\mathbf{X}^{{}^{\prime}}\mid\mid\mathbf{X}\right)=\int
P\left(\mathbf{X}^{{}^{\prime}}\right)\ln\left(\frac{P\left(\mathbf{X}^{{}^{\prime}}\right)}{P\left(\mathbf{X}\right)}\right)d\mathbf{X}^{{}^{\prime}}\\\
&=\int
P\left(\mathbf{X}^{{}^{\prime}}\right)\ln\left(P\left(\mathbf{X}^{{}^{\prime}}\right)\right)d\mathbf{X}^{{}^{\prime}}-\int
P\left(\mathbf{X}^{{}^{\prime}}\right)\ln\left(P\left(\mathbf{X}\right)\right)d\mathbf{X}^{{}^{\prime}}\\\
&=\int
P\left(\mathbf{X}^{{}^{\prime}}\right)\ln\left(P\left(\mathbf{X}^{{}^{\prime}}\right)\right)d\mathbf{X}^{{}^{\prime}}-\int
P\left(\mathbf{X}^{{}^{\prime}}\right)\ln\left(P\left(\mathbf{X}^{{}^{\prime}}-\Delta\mathbf{X}^{{}^{\prime}}\right)\right)d\mathbf{X}^{{}^{\prime}}\\\
&\approx\frac{1}{2}\left(\Delta\mathbf{X}^{{}^{\prime}}\right)^{2}\int
P\left(\mathbf{X}^{{}^{\prime}}\right)\left(\frac{\partial}{\partial\mathbf{X}^{{}^{\prime}}}\ln\left(P\left(\mathbf{X}^{{}^{\prime}}\right)\right)\right)^{2}d\mathbf{X}^{{}^{\prime}}\\\
&\approx\frac{1}{2}\left(\Delta\mathbf{X}^{{}^{\prime}}\right)^{2}\mathbf{\mathcal{F}}\left(\mathbf{X}^{{}^{\prime}}\right)\\\
\end{split}$ (2)

Here $\mathbf{\mathcal{F}}\left(\cdot\right)$ is noted as the Fisher
information metric. Using the relations eq. 1 and the eq. (LABEL:fisher)
provides a connection between the Fisher information of the cell internal
state and the mutual information with the cellular microenvironment:

$\mathbf{\mathcal{F}}\left(\mathbf{X^{\prime}}\right)=\frac{2\tilde{\beta}}{\Delta\mathbf{X}^{2}}I\left(\mathbf{Y},\mathbf{X}\right)$
(3)

The latter formula implies that the fidelity of the future cell’s internal
state is proportional to the available information in the microenvironment.

###  2.2 Continuous time dynamics

Now, we further assume a very short decision time for the internal variable
evolution $\tau\ll 1$. Along with the Bayesian learning, we assume that the
microenvironmental distribution is a quasi-steady state and therefore we focus
only on the dynamics of the internal variable pdf
$P\left(\mathbf{X^{\prime}})=P(\mathbf{X}+\Delta\mathbf{X},t+\tau\right)$.
Using the Taylor series expansion of two variables, we write

$\begin{split}&P\left(\mathbf{X}+\Delta\mathbf{X},t+\tau\right)=\frac{P\left(\mathbf{Y}\mid\mathbf{X,t}\right)P\left(\mathbf{X},t\right)}{P\left(\mathbf{Y}\right)},\\\
&\implies
P\left(\mathbf{X},t\right)+\Delta\mathbf{X}\nabla_{\mathbf{X}}P\left(\mathbf{X},t\right)+\tau\frac{\partial
P\left(\mathbf{X},t\right)}{\partial
t}+\mathcal{O}(\tau^{2},\Delta\mathbf{X}^{2})=\frac{P\left(\mathbf{Y}\mid\mathbf{X},t\right)P\left(\mathbf{X},t\right)}{P\left(\mathbf{Y}\right)},\\\
&\implies\frac{\partial P\left(\mathbf{X},t\right)}{\partial
t}\approx-\frac{\Delta\mathbf{X}}{\tau}\nabla_{\mathbf{X}}P\left(\mathbf{X},t\right)-\frac{1}{\tau}\left(1-\frac{P\left(\mathbf{Y}\mid\mathbf{X},t\right)}{P\left(\mathbf{Y}\right)}\right)P\left(\mathbf{X},t\right)\\\
\end{split}$ (4)

The term
$\frac{P\left(\mathbf{Y}\mid\mathbf{X},t\right)}{P\left(\mathbf{Y}\right)}$ is
the information flow due to cell sensing (empirical likelihood), behaves like
a drift term. The time-dependent solution of eq. (LABEL:taylor) follows as

$P\left(\mathbf{X},t\right)\propto\left(t-\beta\tau\mathbf{X}\right)e^{-\beta\int^{\mathbf{X}}\left(1-\frac{P\left(\mathbf{Y}\mid\left(t+\beta\tau(\tilde{\mathbf{X}}-\mathbf{X})\right),t\right)}{P\left(\mathbf{Y},\left(t+\beta\tau(\tilde{\mathbf{X}}-\mathbf{X})\right)\right)}\right)d\tilde{\mathbf{X}}}$
(5)

Here, we interpret the factor $\beta=\Delta\mathbf{X}^{-1}$ as the sensitivity
of the cell sensing process. Now, the eq. (LABEL:taylor) reaches a steady
state only when the cell senses perfect the microenvironment, i.e.
$P\left(\mathbf{Y}\mid\mathbf{X},t\right)$ is equal as
$P\left(\mathbf{Y}\right)$. The steady solution of the evolution of
probability distribution helps us to understand how it evolved over a long
time which can tell us how the internal variables of cells settle. So,
$\frac{\partial P\left(\mathbf{X},t\right)}{\partial t}=0$, the
Eq.(LABEL:taylor) can be further written as

$\nabla_{\mathbf{X}}P\left(\mathbf{X},t\right)=-\beta\left(1-\frac{P\left(\mathbf{Y}\mid\mathbf{X}\right)}{P\left(\mathbf{Y}\right)}\right)P\left(\mathbf{X}\right)$
(6)

Now, an exact solution of the steady-state probability distribution reads:

$P\left(\mathbf{X}\right)\propto
e^{\int^{\mathbf{X}}{-\beta\left(1-e^{i\left(\mathbf{Y}:\tilde{\mathbf{X}}\right)}\right)}d\tilde{\mathbf{X}}}$
(7)

The term
$i\left(\mathbf{Y}:\mathbf{X}\right):=\ln\frac{P\left(\mathbf{Y}\mid\mathbf{X}\right)}{P\left(\mathbf{Y}\right)}$
is the pointwise mutual information. Assuming our cell sensing is close to
equilibrium, then the $i\left(\mathbf{Y}:\mathbf{X}\right)\ll 1$ which allows
us to simplify the Eq. (7) into:

$\begin{split}&P\left(\mathbf{X}\right)=\frac{e^{\int^{\mathbf{X}}{-\beta\left(1-e^{i\left(\mathbf{Y}:\tilde{\mathbf{X}}\right)}\right)}d\tilde{\mathbf{X}}}}{Z}\overset{i\left(\mathbf{Y}:\mathbf{X}\right)\rightarrow
0}{=\joinrel=}\frac{e^{\int^{\mathbf{X}}{\beta\hskip
1.42262pti\left(\mathbf{Y}:\tilde{\mathbf{X}}\right)}d\tilde{\mathbf{X}}}}{Z}\\\
&=\frac{e^{\beta\int^{\mathbf{X}}{i\left(\mathbf{Y}:\tilde{\mathbf{X}}\right)d\tilde{\mathbf{X}}}\int{P(\mathbf{Y}|\tilde{\mathbf{X}})d\mathbf{Y}}}}{Z}\\\
&=\frac{e^{-\beta\int^{\mathbf{X}}d\tilde{\mathbf{X}}S(\mathbf{Y}|\mathbf{X}=\tilde{\mathbf{X}})}}{Z},\end{split}$
(8)

where we have used the fact that the
$\int{P(\mathbf{Y}|\tilde{\mathbf{X}})d\mathbf{Y}}=1$ and simplified the term
$\exp(-\beta\int d\tilde{\mathbf{X}}\ln P(\mathbf{Y}))$.

##  3 Connection between Hierarchical Fokker-Planck equation and Bayesian
learning process

In this section, we shall discuss the connection between dissipative dynamics
and Bayesian learning regarding the cell decision-making process. Since cell
decision-making is a stochastic process of the continuous internal variable
$\mathbf{X}$, we can assume the existence of the Fokker-Planck description.
When there exists a timescale separation between two dynamical variables, a
Hierarchical Fokker-Planck equation [32] can be derived. In this section, we
shall show how this formalism can be applied in cell decision-making and also
will show how it helps us to study the origin of biophysical force in terms of
the information-theoretic quantities as shown in Fig.(1).

Figure 1: A schematic picture of cellular decision making in complex
microenvironment through physical forces and through Bayesian learning

Let’s consider $\mathbf{X}$ and $\mathbf{Y}$ to be the internal variables
which evolve in a slow timescale and external variables that are fast and the
corresponding 2-tuple random variables (which evolve over time) as

$\boldsymbol{M}=\begin{pmatrix}\mathbf{M_{1}}\\\
\mathbf{M_{2}}\end{pmatrix}=\begin{pmatrix}\mathbf{\mathbf{X}}\\\
\mathbf{\mathbf{Y}}\end{pmatrix}$ (9)

Now for random variable $\boldsymbol{M}$ one can write the generalized
stochastic differential equation for multiplicative noise processes as

$d\boldsymbol{M}=\boldsymbol{K}\left(\boldsymbol{M},t\right)dt+\mathbf{L}\left(\boldsymbol{M},t\right)d\boldsymbol{W}$
(10)

In this above Eq.(10), we define the drift term $\boldsymbol{K}$, the $L$ that
is a $2\crossproduct 2$ covariance matrix and $d\boldsymbol{W}$ as the Wiener
process [35] which satisfies the mutual independence condition below

$d\boldsymbol{W_{i}}d\boldsymbol{W_{j}}=\delta_{ij}dt$ (11)

The realization of $\mathbf{Y}\equiv M_{1}$ and $\mathbf{X}\equiv M_{2}$ ,
obeys the time-dependent joint probability.
$P\left(\mathbf{X},\mathbf{Y},t\right)$ which satisfies the generalized
Fokker-Planck equation. Now, the generalized Fokker-Planck equation [35, 36,
37] corresponding to the Langevin equation (10) for two-variable homogeneous
processes can be written as

$\frac{\partial P}{\partial t}=-\sum_{p=1}^{,2}\frac{\partial}{\partial
M_{p}}\left(K_{p}P\right)+\sum_{p,q=1}^{2}\frac{\partial^{2}}{\partial
M_{p}\partial M_{q}}\left(\sigma_{qp}P\right)$ (12)

where drift coefficients $K_{p}=K_{p}\left(\mathbf{X},\mathbf{Y},t\right)$ and
diffusion coefficients
$\sigma_{pq}=\sigma_{qp}=\sigma_{pq}\left(\mathbf{X},\mathbf{Y},t\right)$.

The Fokker-Planck equations represent the mesoscopic scale of a dynamical
system [38]. Interestingly, in a large timescale separation at the mesoscopic
level, the degrees of freedom associated with the fast variables depend on
slow variables but not vice versa. Since we have assumed that the
microenvironmental variables $\mathbf{Y}$ evolve at the fastest timescale, it
follows that $K_{1}\equiv K_{1}\left(\mathbf{X},\mathbf{Y}\right)$,
$K_{2}\equiv K_{2}\left(\mathbf{X}\right)$ and
$\sigma_{22}\left(\mathbf{Y},\mathbf{X},t\right)\equiv\sigma_{22}\left(\mathbf{X}\right)$.
This point can be found in the Born–Oppenheimer approximation in quantum
mechanics [39]. To use the separation method adiabatically, we shall
substitute

$P\left(\mathbf{X},\mathbf{Y},t\right)=P\left(\mathbf{Y},t\mid\mathbf{X}\right)P\left(\mathbf{X}\right),$
(13)

where the $P(\mathbf{X})$ is time-invariant relative to the evolution of the
microenvironmental variables. Thus the dynamics of the joint probability
reduces to the dynamics of the fast variable $\mathbf{Y}$ and using the Eq.
(12), we have:

$\begin{split}&\frac{\partial P\left(\mathbf{X},\mathbf{Y},t\right)}{\partial
t}=P\left(\mathbf{X}\right)\frac{\partial
P\left(\mathbf{Y},t\mid\mathbf{X}\right)}{\partial
t}=-P\left(\mathbf{X}\right)\nabla_{\mathbf{Y}}\left(K_{1}\left(\mathbf{Y},\mathbf{X},t\right)P\left(\mathbf{Y},t\mid\mathbf{X}\right)\right)\\\
&-\nabla_{\mathbf{X}}\left(K_{2}\left(\mathbf{X}\right)P\left(\mathbf{Y},t\mid\mathbf{X}\right)P\left(\mathbf{X}\right)\right)\\\
&+P\left(\mathbf{X}\right)\nabla^{2}_{\mathbf{Y}}\left(\sigma_{11}\left(\mathbf{Y},\mathbf{X},t\right)P\left(\mathbf{Y},t\mid\mathbf{X}\right)\right)\\\
&+2\nabla_{\mathbf{X}}\left[P\left(\mathbf{X}\right)\nabla_{\mathbf{Y}}\left(\sigma_{12}\left(\mathbf{Y},\mathbf{X},t\right)\right)P\left(\mathbf{Y},t\mid\mathbf{X}\right)\right]\\\
&+\nabla^{2}_{\mathbf{X}}\left(\sigma_{22}\left(\mathbf{X}\right)P\left(\mathbf{X}\right)P\left(\mathbf{Y},t\mid\mathbf{X}\right)\right)\end{split}$
(14)

From this point, the equations for the fast degree of freedom and the others
(slow degree of freedom and coupling between them) are derived respectively as
follows:

$\frac{\partial P\left(\mathbf{Y},t\mid\mathbf{X}\right)}{\partial
t}=-\nabla_{\mathbf{Y}}\left(K_{1}\left(\mathbf{Y},\mathbf{X},t\right)P\left(\mathbf{Y},t\mid\mathbf{X}\right)\right)+\nabla^{2}_{\mathbf{Y}}\left(\sigma_{11}\left(\mathbf{Y},\mathbf{X},t\right)P\left(\mathbf{Y},t\mid\mathbf{X}\right)\right),$
(15)
$\begin{split}&\nabla_{\mathbf{X}}\left(K_{2}\left(\mathbf{X}\right)P\left(\mathbf{Y},t\mid\mathbf{X}\right)P\left(\mathbf{X}\right)\right)+2\nabla_{\mathbf{X}}\left[P\left(\mathbf{X}\right)\nabla_{\mathbf{Y}}\left(\sigma_{12}\left(\mathbf{Y},\mathbf{X},t\right)\right)P\left(\mathbf{Y},t\mid\mathbf{X}\right)\right]\\\
&+\nabla^{2}_{\mathbf{X}}\left(\sigma_{22}\left(\mathbf{X}\right)P\left(\mathbf{X}\right)P\left(\mathbf{Y},t\mid\mathbf{X}\right)\right)=0\end{split}$
(16)

From Eq. (16), if we integrate once over $\mathbf{X}$ it follows

$\begin{split}&-K_{2}\left(\mathbf{X}\right)P\left(\mathbf{Y},t\mid\mathbf{X}\right)P\left(\mathbf{X}\right)+2P\left(\mathbf{X}\right)\nabla_{\mathbf{Y}}\left(\sigma_{12}\left(\mathbf{Y},\mathbf{X},t\right)P\left(\mathbf{Y},t\mid\mathbf{X}\right)\right)\\\
&+\nabla_{\mathbf{X}}\left(\sigma_{22}\left(\mathbf{X}\right)P\left(\mathbf{Y},t\mid\mathbf{X}\right)P\left(\mathbf{X}\right)\right)=0,\\\
\end{split}$ (17)

and working further on the equations

$\begin{split}&-K_{2}\left(\mathbf{X}\right)P\left(\mathbf{Y},t\mid\mathbf{X}\right)P\left(\mathbf{X}\right)\\\
&+2P\left(\mathbf{X}\right)\nabla_{\mathbf{Y}}\left(\sigma_{12}\left(\mathbf{Y},\mathbf{X},t\right)\right)P\left(\mathbf{Y},t\mid\mathbf{X}\right)+2P\left(\mathbf{X}\right)\sigma_{12}\left(\mathbf{Y},\mathbf{X},t\right)\nabla_{\mathbf{Y}}\left(P\left(\mathbf{Y},t\mid\mathbf{X}\right)\right)\\\
&+\nabla_{\mathbf{X}}\left(\sigma_{22}\left(\mathbf{X}\right)\right)P\left(\mathbf{Y},t\mid\mathbf{X}\right)P\left(\mathbf{X}\right)+\sigma_{22}\left(\mathbf{X}\right)P\left(\mathbf{X}\right)\nabla_{\mathbf{X}}\left(P\left(\mathbf{Y},t\mid\mathbf{X}\right)\right)\\\
&+\sigma_{22}\left(\mathbf{X}\right)P\left(\mathbf{Y},t\mid\mathbf{X}\right)\nabla_{\mathbf{X}}\left(P\left(\mathbf{X}\right)\right)=0.\\\
\end{split}$ (18)

To isolate the slow degree of freedom, we further separate Eq. (18) as
follows:

$-\left(K_{2}\left(\mathbf{X}\right)-\nabla_{\mathbf{X}}\left(\sigma_{22}\left(\mathbf{X}\right)\right)\right)P\left(\mathbf{X}\right)+\sigma_{22}\left(\mathbf{X}\right)\nabla_{\mathbf{X}}\left(P\left(\mathbf{X}\right)\right)=0,$
(19)
$2\nabla_{\mathbf{Y}}\left(\sigma_{12}\left(\mathbf{Y},\mathbf{X},t\right)P\left(\mathbf{Y},t\mid\mathbf{X}\right)\right)+\sigma_{22}\left(\mathbf{X}\right)\nabla_{\mathbf{Y}}\left(P\left(\mathbf{Y},t\mid\mathbf{X}\right)\right)=0,$
(20)

which are the equations for the slow degree of freedom and the coupling,
respectively. Thus, Eqs. (15), (19) and (20) are the ones to be analyzed. The
general solution of Eq. (19) is

$P\left(\mathbf{X}\right)=f_{0}\exp{\int^{\mathbf{X}}d\tilde{\mathbf{X}}\frac{K_{2}\left(\tilde{\mathbf{X}}\right)}{\sigma_{22}\left(\tilde{\mathbf{X}}\right)}-\ln\sigma_{22}\left(\mathbf{X}\right)}.$
(21)

where $f_{0}$ is a positive constant. If we have information about the drift
term $K_{2}\left(\mathbf{X}\right)$ and diffusion coefficient
$\sigma_{22}\left(\mathbf{X}\right)$, we can easily calculate the probability
distribution of the internal variables from Eq.(21), which is independent of
the fast variable.

In the section of Bayesian learning, we have calculated the steady state
distribution $P\left(\mathbf{X}\right)$ that depends on the mutual information
between internal and external variables. Using this, we will establish the
relationship among the microenvironmental entropy
$S\left(\mathbf{Y}|\mathbf{X}=\mathbf{x}\right)$ and the physical force of
internal states $K_{2}\left(\mathbf{X}\right)$, and corresponding diffusion
present in the system $\sigma_{22}$. Comparing the Eq. (8) and Eq. (21) one
can get

$\begin{split}&P\left(\mathbf{X}\right)=\frac{e^{-\beta\int^{\mathbf{X}}d\tilde{\mathbf{X}}S(\mathbf{Y}|\mathbf{X}=\tilde{\mathbf{X}})}}{Z}=f_{0}\exp{\int^{\mathbf{X}}d\tilde{\mathbf{X}}\frac{K_{2}\left(\tilde{\mathbf{X}}\right)}{\sigma_{22}\left(\tilde{\mathbf{X}}\right)}-\ln\sigma_{22}\left(\mathbf{X}\right)},\\\
&\frac{e^{-\beta\int^{\mathbf{X}}d\tilde{\mathbf{X}}S(\mathbf{Y}|\mathbf{X}=\tilde{\mathbf{X}})}}{Z}=\tilde{f}\exp{\frac{1}{\sigma_{22}}\int^{\mathbf{X}}K_{2}\left(\tilde{\mathbf{X}}\right)d\tilde{\mathbf{X}}},\\\
&-\beta\int^{\mathbf{X}}d\tilde{\mathbf{X}}S(\mathbf{Y}|\mathbf{X}=\tilde{\mathbf{X}})=\ln\left[\tilde{f}Z\right]+\frac{1}{\sigma_{22}}\int^{\mathbf{X}}K_{2}\left(\tilde{\mathbf{X}}\right)d\tilde{\mathbf{X}},\\\
&-\beta
S\left(\mathbf{Y}|\mathbf{X}=\mathbf{x}\right)=\frac{1}{\sigma_{22}}K_{2}\left({\mathbf{X}}\right)\\\
&\implies
K_{2}\left({\mathbf{X}}\right)=-\beta\sigma_{22}S\left(\mathbf{Y}|\mathbf{X}=\mathbf{x}\right).\\\
\end{split}$ (22)

In this above Eq. (LABEL:mutrel) $\tilde{f}$ is defined as
$\frac{f_{0}}{\sigma_{22}}$ and the diffusion coefficient
$\sigma_{22}\left({\mathbf{X}}\right)$ in Eq. (LABEL:mutrel) is constant i.e.,
$\sigma_{22}\left(\mathbf{X}\right)=\sigma_{22}$. In the above relation, we
can directly see how the microenvironmental entropy and the drift force have a
one-to-one relation.

##  4 Implications of cell sensing activity

Cell sensing is usually defined as a process where cells communicate with the
external environment based on their internal regulatory network of signalling
molecules. In the context of Bayesian learning cells, the cell sensing
distribution $P(\mathbf{Y}|\mathbf{X})$ plays a central role. The problem is
that the regulation between a particular sensing molecule and the set of
microenvironmental variables can be pretty complex [40]. Without loss of
generality, we constrain ourselves to one-dimensional internal and external
variables. Let’s consider the microenvironment $Y$ is sensed by the internal
state $X$ as

$Y_{X}=Y\mid X=F\big{(}X,\langle Y^{n}\rangle\big{)}.$ (23)

Here, we assume that the cell sensing function $F(\cdot)$ also depends on
moments of the microenvironmental variable. Now, if we do a Taylor series
expansion around the mean value of the internal state $\bar{X}$ in Eq.(23):

$\begin{split}&Y_{X}=F(\bar{X})+\Big{|}\frac{\partial}{\partial
X}F(\bar{X})\Big{|}\left(X-\bar{X}\right)\\\
&Y_{X}-\bar{Y}=F(\bar{X})-\bar{Y}+\Big{|}\frac{\partial}{\partial
X}F(\bar{X})\Big{|}\left(X-\bar{X}\right)\\\ &\sigma^{2}_{Y\mid
X}(x)=\big{\langle}b+g(x-\bar{X})\big{\rangle}_{P(Y)}^{2}\\\ \end{split}$ (24)

Here, we define the bias term $b=F(\bar{X})-\bar{Y}$ and the linear sensing
response to microenvironmental changes $Y$ defined by
$g=\mid\frac{\partial}{\partial X}F(\bar{X})\mid$. The biological relevance of
this linear sensing function can be found in the classical receptor-ligand
models [41]. In particular, let us assume that the sensed environment variable
$Y|X$ is the ligand-receptor complex and the variable $X$ corresponds to the
receptor density. If $g$ is a first order Hill function for the first moment
of $Y$, who in this context is the ligand concentration, and if
$F(\bar{X})=0$, then first Eq. (23) corresponds to the textbook steady state
of the complex formation [41].

Moreover, we consider the microenvironmental distribution as Gaussian, where
the entropy of the microenvironment, conditioned by the corresponding internal
states, can be written as

$S(Y\mid X=x)=\frac{1}{2}\ln\left(2\pi e\sigma^{2}_{Y\mid X}(x)\right).$ (25)

Now, using the above expression of microenvironmental conditional entropy one
can calculate the steady state of cellular internal variables from Bayesian
learning using the eq.(8). It can be written as

$\begin{split}&P\left(X\right)\propto e^{-\beta\int^{X}{S\left(Y\mid
X=\tilde{X}\right)d\tilde{X}}}\\\
&=e^{-\beta\int^{X}{\ln{\left(b+g(\tilde{X}-\bar{X})\right)}d\tilde{X}}}\\\
\end{split}$ (26)

Interestingly, we have two cases to study the steady-state distribution of the
cellular internal states: $(\mathbf{I})$ when the response of $X$ to
microenvironmental changes is negligible and $(\mathbf{II})$ when there exists
a finite correlation value between internal cellular state and
microenvironmental state, which follows as

$\begin{split}&P\left(X\right)=C_{1}e^{-\bar{\beta}X},\hskip 28.45274ptg\ll
1\\\
&P\left(X\right)=C_{2}\left(b+g\left(X-\bar{X}\right)\right)^{\beta\left(X-\bar{X}+\frac{b}{g}\right)}e^{-\beta
X},\hskip 28.45274ptg=\mathcal{O}(1)\\\ \end{split}$ (27)

Here $C_{0}$ and $C_{1}$ are normalization constants of corresponding
probability distributions and $\bar{\beta}$ is defined as $\beta\ln{b}$. In
case $(\mathbf{I})$ i.e., when $g$ is equal to $0$ the steady state
distribution of internal variables converges to an exponential distribution.
Please note that the sensor OFF probability distribution makes sense only for
$\bar{\beta}>0$. In the ON case, when the linear response $g$ is finite and
$\beta<0$ the expression of the steady state has a unimodal. Interestingly,
for and $\beta>0$ and for a finite range of $X$ values the distribution is
bimodal with the highest probability density around the boundaries of the
domain. In a nutshell, the above expression of the internal state shows how an
ON-OFF switching case can happen when the environment correlates with the cell
and as a response cell senses the microenvironment changing its phenotype
which confirms the existence of monostable-bistable regime as shown in
fig.(2).

Figure 2: Plot of the normalized steady-state probability distribution of
cellular phenotypes for both cases $(\mathbf{I})$ $g=0$ and $(\mathbf{II})$
$g=1$ with different values of $\beta$. $b$ and $\bar{X}$ parameter is kept at
$2$.

##  5 Bayesian learning minimizes the microenvironmental entropy in time

Recently, we have postulated the Least Environmental Uncertainty Principle
(LEUP) for the decision-making of cells in their multicellular context [21,
22]. The main premise of LEUP is that the microenvironmental
entropy/uncertainty is reducing in time. Here, we have hypothesized that cells
use Bayesian learning to infer their internal states from microenvironmental
information. In particular, we have previously shown that
$\frac{dS(\mathbf{Y}|\mathbf{X})}{dt}\leq 0$ [22], which is the case in the
Bayesian learning case. To illustrate this let’s focus on the Gaussian 1D case
of the previous section. Averaging the Eq. (24) for the distribution $p(X,Y)$,
we can obtain the following:

$\sigma^{2}_{Y\mid X}=b^{2}+g^{2}\sigma_{x}^{2}.$ (28)

One can show that the linear response term is proportional to the covariance
of the internal and external variables, i.e. $g\propto cov(X,Y)$. As the
Bayesian learning is reaching equilibrium, according to Eq. (1) the covariance
approaches zero and consequently

$\sigma^{2}_{Y\mid X}\xrightarrow{t\rightarrow\infty}b^{2}.$

Please note that we still assume that the microenvironmental pdf is in a
quasi-steady state due to the time scale separation. The latter implies that
the variance of $Y|X$ is monotonically decreasing and therefore $S(Y|X)$ is
also a decaying function in time. Therefore, we can postulate that Bayesian
learning is compatible with the LEUP idea.

Mathematically speaking, the original LEUP formulation was employing an
entropy maximization principle, where one can calculate the distribution of
cell internal states using as a constraint the mutual information between
local microenvironment variables and internal variables. The corresponding
variational formulation reads:

$\begin{split}\frac{\delta}{\delta
P\left({\mathbf{X}}\right)}&\Bigg{\\{}S\left({\mathbf{X}}\right)+\beta\bigg{[}\int
d{\mathbf{X}}P\left({\mathbf{X}}\right)\int
d{\mathbf{Y}}P(\mathbf{Y}\mid\mathbf{X})\mathfrak{i}\left({\mathbf{Y}}:{\mathbf{X}}\right)-\bar{I}\left(\mathbf{Y},\mathbf{X}\right)\bigg{]}\\\
&-\lambda\bigg{[}\int
P\left({\mathbf{X}}\right)\mathrm{d}{\mathbf{X}}-1\bigg{]}\Bigg{\\}}=0,\end{split}$
(29)

Here $\delta/\delta P\left({\mathbf{X}}\right)$ is the functional derivative
with respect to the internal states. The two Lagrange multipliers in Eq. (29),
i.e., $\beta$ and $\lambda$ are associated with the steady-state value of the
mutual information $\bar{I}\left(\mathbf{Y},\mathbf{X}\right)$ and the
normalization constant of the probability distribution. The constraint or the
partial information about the internal and external variables is written in
terms of the statistical observable. Solving Eq. (29), we can find Gibbs
probability distribution:

$P\left({\mathbf{X}}\right)=\frac{\exp{\beta\hskip
1.42262ptD\left({\mathbf{Y}}\mid\mathbf{X}=\mathbf{x}||{\mathbf{Y}}\right)}}{Z}=\frac{e^{-\beta
S\left({\mathbf{Y}}\mid{\mathbf{X}=\mathbf{x}}\right)}}{Z^{{}^{\prime}}}.$
(30)

Here $Z=\int e^{\beta\hskip
1.42262ptD\left(\mathbf{Y}|\mathbf{X}=\mathbf{x}||{\mathbf{Y}}\right)}\mathrm{d}{\mathbf{X}}$
and $Z^{{}^{\prime}}=\int e^{-\beta
S\left({\mathbf{Y}}\mid\mathbf{X}=\mathbf{x}\right)}\mathrm{d}{\mathbf{X}}$
are the normalization constants. In the left-hand side of Eq.(30), $\exp{\beta
S\left(\mathbf{Y}\right)}$ has been simplified from the numerator and the
normalization factor.

Interestingly, it can coincide with the Bayesian learning context as a special
case where the $i\left({\mathbf{Y}}:{\mathbf{X}}\right)\rightarrow 0$. Using
Eq. (8) in a finite domain $\mathbf{X}\in\Omega$ and the mean value theorem
for integration, there exists a value $\mathbf{x}$ such that:

$P\left(\mathbf{X}=\mathbf{x}\right)=\frac{e^{-\beta
S(\mathbf{Y}|\mathbf{X}=\mathbf{x})}}{Z^{\prime}},$ (31)

where the volume $|\Omega|$ is simplified. Interestingly, the maximum entropy
distribution (30) and the Bayesian learning steady state distribution (8)
coincide when the random variable $\mathbf{X}$ takes values in the vicinity of
$\mathbf{x}$.

##  6 Discussion

In this paper, we elaborated on the idea of cellular decision-making in terms
of the idea of Bayesian learning. We assumed the existence of a time-scale
separation between environmental and internal variables and subsequently
derived a stochastic description for the temporal evolution of the
corresponding dynamics. In this context, we have studied the impact of cell
sensing on the internal state distribution and the corresponding
microenvironmental entropy evolution.

An interesting finding is the steady state distributions of the internal state
depending on the state of the cell sensor. When the cell does not sense its
microenvironment the internal state distribution is exponentially decaying.
When it is ON then a unimodal distribution occurs which implies a precise
number of e.g. receptors is most likely to be expressed according to a certain
stimulus. The former can be viewed as the physiological modus operandi of the
cell. However, when the sensitivity $\beta$ changes sign then the probability
mass is concentrated to the extreme values of the internal state value which
can be mediated by a potential bistability mechanism. The latter can be
relevant in the context of cancer where a bimodal gene expression can be a
prognostic biomarker [42, 43]. However, such bimodality is not associated only
with pathologies, since it can be occurred in healthy immune cells [44]. It
would be interesting to explore if the sensing activity is a plausible
mechanism of explaining transitions from unimodality to bimodality.

One important point of interest is the range of validity of regarding the
timescale separation between cell decision and the cell’s microenvironmental
variables. In particular, we have assumed that internal state characteristic
time is slower than the microenvironmental one, which can be true for decision
timescales related to the cell cycle duration. Sometimes cell decisions may
seem to be happening within one cell cycle, but the underlying molecular
expressions may evolve even over many cell cycles [33, 34]. During cell cycle
time, we can safely assume that external variables such as chemical signal
concentrations or migrating cells will be in a quasi-equilibrium state.
However, for cell decisions with shorter timescales, such as migration-related
processes which are at the order of one hour, this assumption needs to be
relaxed. In the latter case, the discrete-time dynamics presented in Sec. 2
are still valid.

Here, we assumed that the fast time scale environmental variables can be
influenced by the current state of cellular internal variables. However, we
did not consider the influence of the past time states. This would imply non-
Markov dynamics for internal cellular state evolution. It would be interesting
to study how this assumption could impact the information flow dynamics
between environmental states and cellular internal variables.

The outlined theory is related to single-cell decision-making. Our ultimate
goal is to understand how Bayesian learning is impacting the collective
behaviour of a multicellular system. An agent-based model driven by Bayesian
learning dynamics could be used to analyze the collective dynamics as in [22].
Interestingly, we expect a Bayesian learning multicellular theory to produce
similar results to the rattling interactions introduced in [45]. Similarly, in
rattling dynamics, an approximation of the mutual information between
neighbouring individuals is minimized leading to the emergence of a self-
organized active collective state.

Regarding cell sensing, we took an agnostic approach where a generic function
was assumed. Linearizing the sensing function lead to steady-state dynamics
which could be seen in the ligand-receptor dynamics[46], e.g. by assuming our
sensed environment variable $Y|X$ is the ligand-receptor complex and the
variable $X$ the receptors. It will be alluring to further investigate the
non-linear relationship between internal and external variables which means
considering a few more terms in the Taylor series expansion of conditional
variance to simulate a greater variety of biological sensing scenarios.

Assuming Bayesian learning/LEUP as a principle of cell decision-making, we can
bypass the need for a detailed understanding of the underlying biophysical
processes. Here we have shown that even by using an unknown cell sensing
function, we can accurately infer the state of the cell with a minimal number
of parameters. Building on these concepts, we can create theories and
predictive tools that do not require comprehensive knowledge of the underlying
regulatory mechanisms.

## Acknowledgements

AB and HH thank Prof. Sumiyoshi Abe for the useful discussions. AB thanks the
University of Montreal. HH and AB would like to thank Volkswagenstiftung for
its support of the "Life?" program (96732). HH has received funding from the
Bundes Min-isteriums für Bildung und Forschung under grant agreement No.
031L0237C (MiEDGE project/ERACOSYSMED). Finally, H.H. acknowledges the support
of the FSU grant 2021-2023 grant from Khalifa University.

## References

  * [1] Herbert A. Simon  “The new science of management decision.”  New York: Harper & Brothers, 1960  URL: http://content.apa.org/books/13978-000
  * [2] Clive G. Bowsher and Peter S. Swain  “Environmental sensing, information transfer, and cellular decision-making”  In _Curr. Opin. Biotechnol._ 28 Elsevier Ltd, 2014, pp. 149–155  DOI: 10.1016/j.copbio.2014.04.010
  * [3] B. Alberts  “Molecular Biology of the Cell”  New York: W. W, 2015
  * [4] L. Handly, Jason Yao and Roy Wollman  “Signal Transduction at the Single-Cell Level: Approaches to Study the Dynamic Nature of Signaling Networks”  In _Journal of Molecular Biology_ 428.19, 2016, pp. 3669–3682  URL: http://www.sciencedirect.com/science/article/pii/S0022283616302546
  * [5] Laura Prochazka, Yaakov Benenson and Peter W. Zandstra  “Synthetic gene circuits and cellular decision-making in human pluripotent stem cells”  In _Curr. Opin. Syst. Biol._ 5.September Elsevier Ltd, 2017, pp. 93–103  DOI: 10.1016/j.coisb.2017.09.003
  * [6] Santhosh Palani and Casim A. Sarkar  “Integrating Extrinsic and Intrinsic Cues into a Minimal Model of Lineage Commitment for Hematopoietic Progenitors”  In _PLOS Computational Biology_ 5.9 Public Library of Science, 2009, pp. e1000518  DOI: 10.1371/journal.pcbi.1000518
  * [7] Gábor Balázsi, Alexander Oudenaarden and James J. Collins  “Cellular decision making and biological noise: from microbes to mammals”  In _Cell_ 144.6, 2011, pp. 910–925  DOI: 10.1016/j.cell.2011.01.030
  * [8] Joseph R. Casey, Sergio Grinstein and John Orlowski  “Sensors and regulators of intracellular pH”  In _Nature Reviews Molecular Cell Biology_ 11.1 Nature Publishing Group, 2010, pp. 50–61  DOI: 10.1038/nrm2820
  * [9] Ryuichi Tanimoto et al.  “Detection of Temperature Difference in Neuronal Cells”  In _Scientific Reports_ 6.March Nature Publishing Group, 2016, pp. 1–10  DOI: 10.1038/srep22071
  * [10] Yanina Alvarez and Michael Smutny  “Emerging Role of Mechanical Forces in Cell Fate Acquisition”  In _Frontiers in Cell and Developmental Biology_ 10.May, 2022, pp. 1–7  DOI: 10.3389/fcell.2022.864522
  * [11] Dennis E. Discher, Paul Janmey and Yu Li Wang  “Tissue cells feel and respond to the stiffness of their substrate”  In _Science_ 310.5751, 2005, pp. 1139–1143  DOI: 10.1126/science.1116995
  * [12] Pierre Henri Puech and Pierre Bongrand  “Mechanotransduction as a major driver of cell behaviour: Mechanisms, and relevance to cell organization and future research”  In _Open Biology_ 11.11, 2021  DOI: 10.1098/rsob.210256
  * [13] Spiros A. Vlahopoulos et al.  “Dynamic aberrant NF-$\kappa$B spurs tumorigenesis: A new model encompassing the microenvironment”  In _Cytokine & Growth Factor Reviews_ 26.4, 2015, pp. 389–403  URL: http://www.sciencedirect.com/science/article/pii/S1359610115000477
  * [14] Kepeng Wang, Sergei I. Grivennikov and Michael Karin  “Implications of anti-cytokine therapy in colorectal cancer and autoimmune diseases”  In _Annals of the Rheumatic Diseases_ 72.suppl 2, 2013, pp. ii100  DOI: 10.1136/annrheumdis-2012-202201
  * [15] Giovanni Solinas et al.  “JNK1 in Hematopoietically Derived Cells Contributes to Diet-Induced Inflammation and Insulin Resistance without Affecting Obesity”  In _Cell Metabolism_ 6.5, 2007, pp. 386–397  URL: http://www.sciencedirect.com/science/article/pii/S1550413107002926
  * [16] James O Berger  “Statistical decision theory and Bayesian analysis”  Springer Science & Business Media, 2013
  * [17] Theodore J Perkins and Peter S Swain  “Strategies for cellular decision-making.”  In _Molecular systems biology_ 5.326 Nature Publishing Group, 2009, pp. 326  DOI: 10.1038/msb.2009.83
  * [18] Simo Särkkä  “Bayesian Filtering and Smoothing”, Institute of Mathematical Statistics Textbooks  Cambridge University Press, 2013  DOI: 10.1017/CBO9781139344203
  * [19] Andreas Mayer, Vijay Balasubramanian, Aleksandra M. Walczak and Thierry Mora  “How a well-adapting immune system remembers”  In _Proceedings of the National Academy of Sciences of the United States of America_ 116.18, 2019, pp. 8815–8823  DOI: 10.1073/pnas.1812810116
  * [20] Andrea Auconi, Maja Novak and Benjamin M. Friedrich  “Gradient sensing in Bayesian chemotaxis”  In _Epl_ 138.1, 2022, pp. 1–16  DOI: 10.1209/0295-5075/ac6620
  * [21] Haralampos Hatzikirou  “Statistical mechanics of cell decision-making: the cell migration force distribution”  In _Journal of the Mechanical Behavior of Materials_ 27.1-2 Berlin, Boston: De Gruyter, 2018, pp. 20180001  URL: https://www.degruyter.com/view/journals/jmbm/27/1-2/article-20180001.xml
  * [22] Arnab Barua, Josue M. Nava-Sedeño and Haralampos Hatzikirou  “A least microenvironmental uncertainty principle (LEUP) as a generative model of collective cell migration mechanisms”  In _bioRxiv_ Cold Spring Harbor Laboratory, 2019  DOI: 10.1101/404889
  * [23] Arnab Barua et al.  “Entropy-driven cell decision-making predicts ‘fluid-to-solid’ transition in multicellular systems”  In _New Journal of Physics_ 22.12 IOP Publishing, 2020  DOI: 10.1088/1367-2630/abcb2e
  * [24] Arnab Barua, Alireza Beygi and Haralampos Hatzikirou  “Close to Optimal Cell Sensing Ensures the Robustness of Tissue Differentiation Process: The Avian Photoreceptor Mosaic Case”  In _Entropy_ 23.7, 2021, pp. 867  DOI: 10.3390/e23070867
  * [25] A.. Fokker  “Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld”  In _Annalen der Physik_ 348.5, 1914, pp. 810–820  DOI: 10.1002/andp.19143480507
  * [26] Leo P. Kadanoff  “Statistical physics: statics, dynamics and renormalization”  Singapore: World Scientific, 2007, pp. 483
  * [27] Till Daniel Frank  “Nonlinear Fokker-Planck equations: fundamentals and applications”, Sringer series in synergetics  Berlin: Springer, 2010, pp. 404
  * [28] Christian Rödenbeck, Christian Beck and Holger Kantz  “Dynamical systems with time scale separation: averaging, stochastic modelling, and central limit theorems”  In _Stochastic Climate Models_ Basel: Birkhäuser Basel, 2001, pp. 189–209
  * [29] G.. Ford, M. Kac and P. Mazur  “Statistical Mechanics of Assemblies of Coupled Oscillators”  In _Journal of Mathematical Physics_ 6.4 American Institute of Physics, 1965, pp. 504–515  DOI: 10.1063/1.1704304
  * [30] P Romanczuk et al.  “Active Brownian particles From individual to collective stochastic dynamics”  In _The European Physical Journal Special Topics_ 202 Springer, 2012, pp. 1–162
  * [31] M. Schienbein and H. Gruler  “Langevin equation, Fokker-Planck equation and cell migration”  In _Bulletin of Mathematical Biology_ 55.3, 1993, pp. 585–608  DOI: 10.1007/BF02460652
  * [32] Sumiyoshi Abe  “Fokker-Planck approach to non-Gaussian normal diffusion: Hierarchical dynamics for diffusing diffusivity”  In _Phys. Rev. E_ 102 American Physical Society, 2020, pp. 042136  DOI: 10.1103/PhysRevE.102.042136
  * [33] Dmitry Nevozhay et al.  “Mapping the environmental fitness landscape of a synthetic gene circuit”  In _PLoS computational biology_ 8.4 Public Library of Science San Francisco, USA, 2012, pp. e1002480
  * [34] Alex Sigal et al.  “Variability and memory of protein levels in human cells”  In _Nature_ 444.7119 Nature Publishing Group, 2006, pp. 643–646
  * [35] N.. Van Kampen and William P. Reinhardt  “<i>Stochastic Processes in Physics and Chemistry</i>”  In _Physics Today_ 36.2, 1983, pp. 78–80  DOI: 10.1063/1.2915501
  * [36] Hannes Risken  “The Fokker-Planck equation: methods of solution and applications” 18, Springer series in synergetics  Berlin: Springer, 1996, pp. 472
  * [37] C.. Gardiner  “Stochastic methods: a handbook for the natural and social sciences”, Springer series in synergetics  Berlin: Springer, 2009, pp. 447
  * [38] Pep Español  “Statistical Mechanics of Coarse-Graining”  In _Novel Methods in Soft Matter Simulations_ Berlin, Heidelberg: Springer Berlin Heidelberg, 2004, pp. 69–115  DOI: 10.1007/978-3-540-39895-0_3
  * [39] M. Born and R. Oppenheimer  “Zur Quantentheorie der Molekeln”  In _Annalen der Physik_ 389.20, 1927, pp. 457–484  DOI: https://doi.org/10.1002/andp.19273892002
  * [40] Christina J. Su et al.  “Ligand-receptor promiscuity enables cellular addressing”  In _Cell Systems_ 13.5 The Authors, 2022, pp. 408–425.e12  DOI: 10.1016/j.cels.2022.03.001
  * [41] Douglas A Lauffenburger and Jennifer Linderman  “Receptors: models for binding, trafficking, and signaling”  Oxford University Press, 1996
  * [42] Josivan Ribeiro Justino et al.  “An integrated approach to identify bimodal genes associated with prognosis in cancer”  In _Genetics and Molecular Biology_ 44.3, 2021  DOI: 10.1590/1678-4685-GMB-2021-0109
  * [43] Laura Moody, Suparna Mantha, Hong Chen and Yuan Xiang Pan  “Computational methods to identify bimodal gene expression and facilitate personalized treatment in cancer patients”  In _Journal of Biomedical Informatics: X_ 1.December 2018 Elsevier Inc., 2019, pp. 100001  DOI: 10.1016/j.yjbinx.2018.100001
  * [44] Alex K. Shalek et al.  “Single-cell transcriptomics reveals bimodality in expression and splicing in immune cells”  In _Nature_ 498.7453 Nature Publishing Group, 2013, pp. 236–240  DOI: 10.1038/nature12172
  * [45] Pavel Chvykov et al.  “Low rattling: A predictive principle for self-organization in active collectives”  In _Science_ 371.6524 American Association for the Advancement of Science, 2021, pp. 90–95  DOI: 10.1126/science.abc6182
  * [46] William Bialek  “Biophysics: Searching for principles”  In _Biophys. Search. Princ._ Princeton University Press, 2012