# On Artificial Life and Emergent Computation in Physical Substrates
††thanks: This work was conducted as part of the SOCRATES project, which is
partially funded by the Norwegian Research Council (NFR) through their
IKTPLUSS research and innovation action on information and communication
technologies under the project agreement 270961.

1st Kristine Heiney  Department of Computer Science
Oslo Metropolitan University
Oslo, Norway
<EMAIL_ADDRESS>2nd Gunnar Tufte  Department of Computer Science
Norwegian University of Science and Technology
Trondheim, Norway
<EMAIL_ADDRESS>3rd Stefano Nichele  Department of Computer Science
Oslo Metropolitan University,
Department of Holistic Systems
Simula Metropolitan
Oslo, Noway
<EMAIL_ADDRESS>
###### Abstract

In living systems, we often see the emergence of the ingredients necessary for
computation—the capacity for information transmission, storage, and
modification—begging the question of how we may exploit or imitate such
biological systems in unconventional computing applications. What can we gain
from artificial life in the advancement of computing technology? Artificial
life provides us with powerful tools for understanding the dynamic behavior of
biological systems and capturing this behavior in manmade substrates. With
this approach, we can move towards a new computing paradigm concerned with
harnessing emergent computation in physical substrates not governed by the
constraints of Moore’s law and ultimately realize massively parallel and
distributed computing technology. In this paper, we argue that the lens of
artificial life offers valuable perspectives for the advancement of high-
performance computing technology. We first present a brief foundational
background on artificial life and some relevant tools that may be applicable
to unconventional computing. Two specific substrates are then discussed in
detail: biological neurons and ensembles of nanomagnets. These substrates are
the focus of the authors’ ongoing work, and they are illustrative of the two
sides of the approach outlined here—the close study of living systems and the
construction of artificial systems to produce life-like behaviors. We conclude
with a philosophical discussion on what we can learn from approaching
computation with the curiosity inherent to the study of artificial life. The
main contribution of this paper is to present the great potential of using
artificial life methodologies to uncover and harness the inherent
computational power of physical substrates toward applications in
unconventional high-performance computing.

###### Index Terms:

bio-inspired computation, biological neural networks, nanomagnetic ensembles,
artificial life, philosophy of computation

##  I Introduction

The field of artificial life concerns itself with how to produce complex
macroscopic behaviors from the interaction of many simple interacting
components. Where biology works to understand existing organisms and the
complicated machineries that underlie observed physiological behaviors using
an approach of deconstruction and element-by-element description, artificial
life seeks to construct systems displaying interesting emergent behaviors by
aggregating many simple objects governed by basic rules [1]. Here, “emergent”
refers to some feature of the entire system that cannot be described by the
constituent parts of the system. For example, the physical concept of pressure
has no meaning when considering only one or a few individual gas molecules; it
is only when a large volume of gas is considered that this characteristic
emerges as a meaningful descriptor of the system. Similarly, the movement and
behavior of a single ant is qualitatively distinct from that of an entire
colony. In many ways, these emergent behaviors may be seen as a form of
computation, with the systems or organisms providing the machinery by which
computations are performed.

The tools of artificial life allow us to capture these emergent behaviors
without explicitly encoding them into the system. Rather, by creating simple
sets of rules to describe the behavior of individual agents within the system
as they move, connect, and interact, complex behaviors emerge of their own
accord. This approach to modeling and engineering dynamical systems can offer
new perspectives on how to perform computation in substrates showing emergent
properties, positioning us to answer the question posed by Langton [2] for
targeted physical systems: under what conditions might the capacity to perform
computation emerge in a physical system? Ongoing work being conducted by the
authors involves the study of two physical substrates—networks of biological
neurons [3, 4] and nanomagnetic ensembles [5]—along with models of these
systems at different levels of abstraction [6], and we argue that this
inquisitive and bottom-up approach to understanding, mimicking, and
constructing dynamical systems will prove fruitful in the advancement of bio-
inspired parallel and distributed computing technology.

Sipper [7] highlights the three cornerstones of this computational paradigm,
which he terms “cellular computing”: simplicity, vast parallelism, and
locality. In this paradigm, computation is performed with a vast number of
very simple fundamental units whose connections are sparse and most often in
the immediate vicinity. Thanks to this local connectivity, these machines thus
perform without any centralized control, and their function is resilient
against faults in the system. Currently, much exploration into cellular
computing is confined to simulation, though the ultimate aim is to construct
actual machines, be they biological [8, 4] or manmade [5], that can realize
this behavior. In comparison with classic von Neumann computing technologies,
machines based in cellular computation principles will be more scalable,
energy efficient, and resilient to failures of single elements [7, 9].

This paper first explores basic concepts and motivations in artificial life
and cellular computing. A selection of approaches to capturing in models
components of biological processes we see in nature are then presented.
Finally, the two abovementioned physical substrates—neuronal networks and
nanomagnetic arrays—are explored in greater detail, and a philosophical
discussion on how the lens of artificial life may inform our investigation of
these substrates is presented.

##  II An Artificial Life Perspective on Complexity

Human curiosity has long driven us to uncover how organisms function—what
drives their behaviors on a macroscopic scale and what microscopic processes
control physiological function. This has in turn driven many to explore means
to recreate lifelike behaviors using mechanical components—these are the
machineries of artificial life.

Early efforts in the field of artificial life focused on imitating behaviors
observed in natural systems. For example, in 1950, William Grey Walter built a
pair of artificial electronic “tortoises” named Elsie and Elmer that moved
toward dim light but away from bright light [Fig. 1; 10]. When he attached
lights to the tortoises themselves, their interaction resulted in complex and
interesting behavior, which he described as giving “an eerie expression of
purposefulness, independence and spontaneity” [10]. Grey Walter took great
delight in learning from these tortoises, named for the Mock Turtle’s teacher
in Alice’s Adventures in Wonderland, who, aptly, was also not exactly a
tortoise: when Alice asked about the choice of name, the Mock Turtle replied,
“We called him Tortoise because he taught us.” Although Grey Walter also
predicted our captivation with the “marvelous processes” of life may ebb as
our aptitude to imitate it grows, that prediction has not borne itself to
fruition. Rather, it seems that the better able we are to capture complex
dynamics in artificial systems, the greater our appreciation for the natural
systems we strive to understand and emulate.

Figure 1: Schematic of Grey Walter’s electronic tortoises, Elsie and Elmer.
Reproduced from Walter [10].

But what makes a system “artificial”? It is not simply the mechanics or the
fact that it is inorganic, nor is it the behavior the system shows, as this
behavior is meant to be as close to that of the natural systems that inspire
its construction. Herbert Simon gave an elegant definition of the artificial
in his book The Sciences of the Artificial [11]:

> “Artificiality connotes perceptual similarity but essential difference,
> resemblance from without rather than within… [W]e may say that the
> artificial object imitates the real by turning the same face to the outer
> system, by adapting, relative to the same goals, to comparable ranges of
> external tasks.”

Simon’s discussion in this section centers on the distinction between the task
fulfilled by a designed system and the capability of the system itself, and
what can be accomplished by artificial and simulated systems. We may create
artificial systems that are perceptually similar to natural systems, systems
that produce precisely the behaviors we wish to see on the scale at which we
wish to see them, despite being inexorably different from within; this,
indeed, is the situation we strive for in the study of artificial life.

What then do we do with such systems once we have captured their behavior? In
some cases, the developed systems represent a means to understanding the
system they mimic; in others, the system is the final product and gives us
some capability that would be otherwise unattainable in conventional man-made
systems. Models and simulations may provide insight into systems that would be
unattainable by mere inspection of the assumptions and laws employed to govern
the simulated system. This is useful when the precise mechanisms governing the
behavior of a system are known at a certain scale but the system dynamics
become more difficult to describe when the system is scaled up by the addition
of more such simple components. Along the same lines, this complex larger-
scale behavior can be useful in decoding the response of the system to
different types of inputs, enabling the use of a dynamical system as a
computational system.

However, the pursuits of researchers interested in artificial life have often
been motivated not by some end goal but simply by curiosity. Much of the
burgeoning research that has recently been conducted in the realm of
artificial intelligence has been preoccupied with building better classifiers,
better computer vision, better autonomous systems—summarily, the optimization
of tools used as a means for computational tasks. Artificial life research is
not necessarily oriented to these aims but serves to explore how we can
harness the dynamics of behaviors of interest we observe in nature. As Langton
put it in his paper on artificial life, artificial intelligence “has focused
primarily on the production of intelligent solutions rather than on the
production of intelligent behavior. There is a world of difference between
these two possible foci” [1]. Thus, in line with this curious, process-driven
approach, this paper does not consider in great detail what we hope to
ultimately do with the behaviors we observe as a motivation for studying
artificial life but instead engages with the behaviors themselves as a point
of interest with the assumption that applications and advancements will fall
out as a natural product of the knowledge we gain with the approaches
described here.

##  III Artificial Life Models and Tools for Unconventional Computing

When considering the use of complex systems for computation, a system should
be capable of three basic operations [2]: the transmission, storage, and
modification of information. To find systems that support these behaviors, we
must be able to construct models of complex systems, develop metrics to
quantify their performance, and search the space of all possible models to
target the desired behaviors.

This section presents a general approach to capturing dynamic systems in
models and characterizing their behavior. The concepts presented here
represent some of the tools that are commonly used to explore the smaller-
scale mechanisms underlying larger-scale complex dynamics. Dynamic systems are
often modeled using an approach in which the system is considered to comprise
a number of discrete components that interact with a set of predefined rules.
Each of these components can be in a given state and can influence or be
influenced by the states of a number of other components to which it is
connected. Some examples of such systems—graphs and cellular automata
(CAs)—will be presented in the following subsections.

The manner in which these dynamic models behave is governed by a set of
parameters and rules defining the connections in the system, the states each
component can take on, and how the state of each component influences the
states of the components to which it is connected. If we wish to find the set
of parameters and rules that produces a desired behavior in the system, we
have a vast space in which to search, and that space is often largely occupied
by many “boring” or undesirable models that fail to produce the behavior we
seek. Thus, algorithms have been developed based on the biological principle
of evolution to allow for a more targeted searching of the space of
possibilities in model design. This approach will be discussed in more detail
at the end of this section.

###  III-A Complex system models

Complex systems are generally modeled using ensembles of discrete elements
that interact with each other using a given set of rules. These elements can
take on discrete or continuous states and can be connected to each other in a
regular, irregular, or random fashion, or they can be agents allowed to move
freely in space. This section presents two illustrative examples of different
types of models used to represent the dynamical behavior of complex systems:
graphs and CAs.

These two types of models were selected for their relevance to the question of
computing in physical substrates. Graphs represent the connections between
elements in a system and how those connections mediate the dynamics of the
system; this type of model is very relevant in the field of neuroscience [12]
and can capture the intricate arrangements of individual cells in a network or
connections among brain regions. CAs, on the other hand, are one of the
simplest representations of a dynamical system, and they offer insight into
the vast range of dynamical behavior that can be achieved by tuning the
parameters of the system, even in the case of a very simple elementary CA.

####  III-A1 Graphs

A graph is composed of nodes, representing the discrete components of the
system, and edges, representing the connections between the nodes. The state
of each node affects the states of all of its neighbors in a manner defined by
a set of rules or equations, much like the flight of a bird is affected by its
fellow flock members. Graphs may also be constructed to have static or dynamic
structures, with connections remaining fixed or evolving over time. The
variation in the states of the nodes is referred to as dynamics on the
network, whereas the change in the connections between the nodes is referred
to as dynamics of the network. Systems that show both types of dynamics are
referred to as dynamic systems with dynamic structures ((DS)2) or adaptive
networks. The brain is a prime example of an adaptive network, with numerous
plasticity mechanisms constantly dynamically tweaking the weights of neural
connections.

A number of useful measures can be extracted from graph maps of a given
structure to give some insight into the dynamics on the network. Examples
highlighted by Sporns et al. [13] as applicable in the study of brain
connectivity are shown in Fig. 2, and further details on graph theoretical
measures can be found in Sporns et al. [12].

Figure 2: Examples of graph theoretical measures commonly used in the study of
complex systems. Reproduced from Sporns et al. [13].

Two network structures that represent opposite extremes are regular and random
networks. Regular networks consist of nodes with the same number of neighbors
and range from strongly regular, where every two adjacent nodes have the same
number $n_{1}$ of neighbors in common and every two non-adjacent nodes have
the same number $n_{2}$ of neighbors in common, to randomly regular, where
every node has the same degree but the connections are randomly distributed.
At the other extreme, random networks have a binomial degree distribution (or
Poisson in the limit of a large number of nodes), meaning the nodes can be
well-described by the average degree.

Many real systems tend to not show regular or random connectivity but lie
somewhere in between these two extremes. Two types of commonly discussed
models representative of real-world behaviors are scale-free and small-world
networks. Scale-free networks have degree distributions that follow a power
law: $P(k)\propto k^{-\gamma}$, where $P(k)$ is the fraction of nodes with
degree $k$ and $\gamma$ is a constant. This means that many nodes have a low
degree and few nodes have a high degree. Although many real networks have been
reported to show scale-free connectivity, it is notoriously difficult to
rigorously confirm power-law scaling from empirical data of finite systems
[14].

Scale-free networks are characterized by the presence of hubs and modularity.
They tend to show local clustering and long-range integration and are robust
against random removal or failure of nodes, as the vast majority of the nodes
have few connections. However, if the hubs in the network are targeted, the
system breaks down [see, e.g., 15]. In comparison to a random network of the
same size and average degree, the mean path length of a scale-free network is
smaller whereas the clustering coefficient is much larger, demonstrating the
modularity of the structure lends itself to efficient network communication.

Graph-based simulations can give insight into the dynamics of a number of
different types of systems and how the interactions of their components can
produce emergent macroscale behavior. Crucially, information flow through a
system of interconnected elements can be readily represented with this
modeling approach.

####  III-A2 Cellular automata

A CA is classically defined as a regular $n$-dimensional lattice structure (or
regularly connected graph) composed of discrete elements called cells that can
take on discrete states. The state of each cell in the network progresses in
discrete time steps according to a lookup table of rules that give the state
at time step $t+1$ based on the states of the cells in the neighborhood at
time step $t$. Although CAs are actually a type of graph, the extra
simplifying constraints placed on them make them a useful case to consider in
the realm of computation.

The binary ($K=2$) one-dimensional CA with a neighborhood of size $N=3$ is one
of the simplest possible dynamical system models to show complex behavior, and
the range of behaviors that can be achieved by this model has been
investigated in great depth. Such a model is an excellent exemplar of a
complex system, as it shows a wide range of dynamic behavior that can serve as
an analogue for the behavior of more complicated systems. In his seminal paper
on the dynamical behavior of simple CAs, Langton [2] explored the different
behaviors that are attainable with this type of system and focused on how
physical systems may show an emergent capacity for computation. This section
will briefly explain the aims and achievements of his study.

Figure 3: Examples of CAs from the four different classes with $K=4$ states
and $N=5$ neighbors. Images reproduced from [2].

As Langton [2] stated, the focus of his paper was to determine “the conditions
under which [the] capacity to support computation itself might emerge in
physical systems” by considering CAs as an exemplary simple model system. To
this end, he qualitatively and quantitatively characterized a number of one-
dimensional CAs with different rules and developed a new quantitative measure,
the $\lambda$ parameter, that can be used to identify the qualitative class of
behavior of a CA. This parameter is defined such that the cases where
$\lambda=0.0$ and $1.0$ correspond to the most homogeneous and heterogeneous
rulesets, respectively.

Wolfram [16] had earlier defined four classes of CA behavior, with classes I
and II corresponding to fixed and periodic behavior, class III showing
aperiodic patterns with no identifiable structure, analogous to chaotic
behavior. Finally, class IV CAs yield “complex patterns of localized
structures,” effectively having “very long transients” [2]. These types of CAs
show rich and interesting patterns of behavior with complex fractal-like
structures emerging and propagating over space and time. Langton’s survey of
the possible rulesets for systems with $K=4$ and $N=5$ revealed an interesting
correspondence between the qualitative behavior observed and the $\lambda$
parameter. Some examples of the observed behavior and the relationship between
$\lambda$ and the class of behavior are shown in Fig. 3.

Shifting his focus to a larger two-dimensional CA, Langton [2] also explored
the relationships among $\lambda$, the average single-cell entropy $H$, and
the mutual information (MI) between a cell and itself at the next time step.
The relationships between these parameters revealed the presence of a sharp
phase transition, corresponding to the transition between classes II and III.
Langton’s results reveal high clustering of many CA rulesets in two distinct
regions corresponding to these classes, with classes I and II occupying a
region of low $H$ and low MI and class III a region of high $H$ and low MI.
However, in the wide gap between these two regions lie a few sparse points
representing the class IV CAs in the transitional regime poised delicately
between order and chaos; here at a point of intermediate entropy, the MI is
maximized in a sharp peak between the low-MI regions on either side.

Langton’s delving into the dynamics at the “edge of chaos” is a remarkably—and
somewhat overwhelmingly—thorough exploration of how dynamics at the phase
transitional regime may give insight into the nature of computation in the
physical world. But what is most crucial to take away here? First, there is
the more conceptual and more challenging lesson: computation can be
accomplished by striking a precarious balance between information storage,
which requires a lowering of the entropy, and information transmission, which
requires raising the entropy. This lesson is applicable to any dynamical
system, not only CAs. At the transitional regime between periodic and chaotic
dynamics, we have the behavior needed for long-term and long-range
correlations, allowing information to propagate arbitrarily far and remain for
arbitrarily long periods of time.

Second, Langton’s parameter $\lambda$ gives us a way to survey the vast space
of all possibilities to hone in on the systems that can show the behaviors we
wish to see in computational systems. Considering how rapidly the number of
possible systems can expand as a modeling framework is adjusted to represent
ever more nuanced features of actual physical systems, this surveying ability
is highly valuable. The following section will also address a more general
approach to surveying the space of possible systems to find desired behavior.

###  III-B Evolutionary Algorithms

As may have been apparent from the explanations of the two example model
systems in the previous section, even relatively small systems with simple
rules governing their local dynamics can often show very complex behavior that
cannot be predicted by examining their structure and rulesets alone; rather,
the system must be run to observe its behavior. For any interesting system,
there will certainly exist a vast number of possible configurations and
rulesets, to the extent that it would not be reasonable to brute force our way
through checking the behavior of each one. Furthermore, as discussed by
Langton [2] and exemplified by his CA survey, the number of configurations
showing interesting behavior becomes vanishingly small with respect to the
space of all possible configurations as the system size scales up. All of
these factors make it challenging to select for systems that show a targeted
type of behavior.

One approach to tackling this issue is the use of evolutionary algorithms,
which take inspiration from the process of evolution by natural selection to
iteratively improve generations of machines to produce the desired behavioral
outcome [1]. In this approach, the rules that govern local interactions are
encoded into a simple representation of a possible system configuration, and
the behavior that is produced when the system is run is evaluated to determine
how well it performs based on a desired metric called the fitness function.
The representation and output behavior are conceptually similar to the
genotype (genetic makeup) and phenotype (observable characteristics) of an
organism.

The process of computational evolution then follows a path analogous to that
in nature. An initial population of individual machines is created and run.
Their fitness is evaluated based on a fitness function quantifying how closely
their behavior resembles the target behavior. The descriptions of those that
perform the best from the population (parent machines) are then used to
generate new descriptions for a new generation of machines (offspring
machines). For this purpose, genetic algorithms are commonly used. These
involve selecting pairs of machines from the parent generation that show high
fitness and performing genetics-inspired operations like crossover and
mutation. This process is iterated over many generations and allows for a more
intelligent search of the space of possible machines. In the realm of
computation in physical substrates, a technique known as evolution-in-materio
is commonly used [17], where genetic algorithms are used to search for
physical signals that can be applied to a physical system to configure its
properties to a desired state, heightening its capacity for computation.

##  IV Working with Physical Substrates

To better understand the behaviors driving complex dynamics in actual physical
systems, data-driven modeling approaches can be employed, where data obtained
from actual physical substrates is obtained and analysis of this data is used
to recapture targeted behaviors in models. In the case of engineerable
substrates, targeted behaviors may even be translatable from natural
biological substrates. Data-driven modeling serves a dual purpose: (1)
providing insight into the behavior of the studied system, including through
simulations of situations that may not be easily achievable experimentally,
and (2) enabling the emulation of natural systems in artificial systems.

Furthermore, increasing attention has turned to the exploitation of physical
nonlinear systems for computation [18]. Current computing substrates are
inflexible and power-hungry; the use of complex nonlinear systems in computing
hardware would open up the possibility for more efficient and powerful
hardware with the capacity to learn and could pave the way for rapid
advancement in artificial intelligence. One example of a computing paradigm
exploiting the nonlinearity of certain systems is reservoir computing [19,
20], one great benefit of which is that the system acting as the reservoir
does not need to be trained or modified; rather, the connections and dynamical
behaviors it shows can be harnessed by finding the appropriate way to encode
inputs to apply to the system and decode the output behavior it produces. This
computing paradigm has been exploited in artificial intelligence applications
[20], and artificial life approaches to computational reservoirs may offer
tools for the advancement of such applications.

This section gives an overview of the dynamics of two physical substrates
investigated by the authors—networks of neurons and nanomagnetic arrays—and
some approaches that have been taken to capture their behavior in
computational models. These two substrates both show the necessary nonlinear
behavior for them to be well-suited for computation.

###  IV-A Neuronal networks

The human brain is arguably the most complicated machine we know of. With very
low power consumption, it can make sense of an impressive range of inputs and
control widely varied bodily responses to these inputs, and it can store vast
amounts of information. Neurons in the brain encode and transmit information
in stereotyped electrical signals called action potentials, or spikes, which
are produced by a carefully synchronized flow of ions across the cell membrane
and can induce spikes in other neurons via connections called synapses.
Although the mechanisms of spike generation, propagation, and transmission are
fairly well characterized at the cellular level—though, admittedly, even on a
single-cell level, what is known is far from simple and there remains much to
be learned—larger-scale behaviors cannot be wholly explained by simply
combining many of these smaller-scale models.

Much research has been devoted to understanding the immense computational
capabilities of the brain and how networks of neurons process, transmit, and
store information, and advances in recording technology and data handling
techniques have opened the door to new lines of investigation that were
previously inaccessible. The electrical behavior of neurons allows them to be
studied at many scales using different techniques, including single-cell
recording by measuring the voltage across the cell membrane and
electroencephalogram (EEG) at the whole-brain scale. To limit the scope of the
discussion here, we focus on the study of the electrophysiological behavior of
neurons at the network or population level using microelectrode array (MEA)
technology [21].

An MEA is a set of electrodes embedded in a substrate, such as glass, on which
neurons can be grown. MEAs enable the long-term nondestructive recording of
populations of neurons as well as controlled network stimulation by electrical
pulses. An example of a 60-electrode MEA is shown in Fig. 4, along with an
example of the voltage signals recorded by the MEA, from which the spiking
behavior of the network can be extracted.

The advent and recent advancement of MEA technology, including accessibility
to commercial recording setups and analysis tools (e.g., Multi Channel Systems
MEA2100 systems and MEAs, along with their corresponding software suite; see
Fig. 4) have made it possible for researchers to perform long-term observation
of the behavior of populations of neurons in vitro. This capacity to record
from whole populations of neurons, both in living animals and in disembodied
cultures, using MEAs and related technology has brought about a shift in focus
from the spiking of single neurons to network-level dynamics and how neuronal
assemblies within the brain can collectively drive the dynamics and function
of the entire system [22]. Indeed, although complex and inarguably worthy of
the attention it has received, the behavior of a single neuron can only tell
us so much when decoupled from its “neighborhood” of other interconnected
neurons.

Figure 4: Microscope image of an MEA (Multi Channel Systems GmbH, Germany)
with a network of neurons cultured on top of it [4] (left) along with a screen
capture from the Multi Channel Suite of software (Multi Channel Systems GmbH,
Germany).

This shift in focus has brought about the necessity to inquire into the
organization of such populations of neurons, raising a number of questions.
What physical feature of the network constitutes a connection between two
neurons? How can we capture these connections in the network through our
observations of it? And what measures can we use to characterize the network
connections?

####  IV-A1 Connectivity in neuronal networks

The organization of networks of neurons is typically described in terms of
three types of connectivity: structural, functional, and effective
connectivity [22]. A structural connection indicates an anatomical feature
that mediates a physical interaction between two neurons, namely a synapse.
Structural connectivity is extracted from imaging data of morphological
features or synaptic markers. Although methods exist for extracting such
features in relatively low-density networks [e.g., 23, 24], where the
morphology of individual neurons can be observed, in higher-density networks
on MEAs, such features can be difficult to extract from images.

Functional connectivity represents the temporal correlation between the
spiking patterns of pairs of neurons obtained using, for example, the cross-
correlation between pairs of spike trains. In this type of connectivity, two
neurons are said to be connected if the spiking of one can be predicted from
the spiking of the other; however, this does not necessarily indicate a causal
relationship between the spiking of the two. The effective connectivity, in
contrast, refers to connections where the activity of one neuron can be said
to directly cause that of another neuron.

In one noteworthy study [23], the structural and functional connectivity were
obtained from imaging and electrophysiological data from a high-density MEA
and then combined to yield a refined functional connectivity map that may be
said to better indicate the actual organization and activity of the network
(Fig. 5). It should be noted that all of these types of connectivity are
subject to change as a result of different types of plasticity mechanisms
operating on a wide range of time scales. This plasticity means that neuronal
systems show both dynamics on the network, in the form of spiking activity
traveling through the network, and dynamics of the network, with the
connections between neurons subject to changing as a result of their activity.

Figure 5: Combined structural and functional connectivity to capture a more
accurate representation of the organization and activity of an in vitro
neuronal network. Adapted from Ullo et al. [23].

####  IV-A2 Connectivity and dynamics: How does information flow?

These types of connectivity give us an idea of what may be considered to
constitute a connection and how to capture such connections—that is, either
visual observation of an anatomical connection or temporal correlations in the
electrophysiological spiking data recorded from pairs or neurons. What
conclusions can then be drawn from these connectivity maps, and what tools can
we use to get there?

A first clear step once the connectivity is obtained is to apply graph theory
measures (see Fig. 2) to characterize the organization of the connectivity
map. These measures can tell us if, for example, the network is modular and
contains many hubs or if it is more randomly connected with many nodes having
roughly the same degree. The brain is known to strike a balance between
functional segregation and functional integration, allowing information to
flow between spatially distant parts of the brain while allowing the
generation of coherent brain states [12]. Mapping the connectivity of neuronal
populations may give us insight into how this balance is struck. Additionally,
as mentioned previously, the connectivity of a network changes over time, both
as a consequence of maturation and in response to stimuli. Connectivity
analysis can provide us a deeper understanding of how neurons organize
themselves over time and what may drive these organizational structures, in
both healthy networks and networks that mimic diseases or other abnormal
states, and can show how the connectivity is affected by the adaptive or
maladaptive responses the networks have to external stimuli or perturbations.

The connectivity of a network is also intimately tied with the dynamics that
happen on the network and the manner in which elements in the network
communicate and process information. One approach to capturing the dynamic
state of a network is to study the distribution of the size of network-wide
cascades of activity called “neuronal avalanches” [25]. It has been theorized
that the brain lies in the critical state, a state analogous to the “edge of
chaos” explored by Langton [2] in which information processing is optimized,
and in this state, the size distribution of neuronal avalanches follows a
power law. Massobrio et al. [26] have shown that a model with scale-free
connectivity is able to reproduce the power-law avalanche scaling we expect to
see in networks at criticality, and Shew et al. [27, 28] have shown optimized
dynamic range, information capacity, and information transmission in networks
showing power-law avalanche scaling. Additionally, preliminary results
indicate it may be possible to manipulate supercritical networks into the
critical state by increasing network inhibition [4], enabling comparative
studies between their behavior in different states and their exploitation as
computational reservoirs. These results demonstrate the link between
connectivity, dynamic state, and information processing in neuronal networks.

Future work into these three perspectives on the dynamical behavior of neurons
may give us invaluable insight into how we can construct self-organizing
systems to show the same kind of capacity for efficient information
processing—how we can organize connections between elements of the system and
construct rules for how they affect each other’s behavior. Building models
like this can in turn give us a deeper understanding of the brain as well, as
we target specific behaviors to emulate and see how simplified models can
produce behaviors analogous to those observed in the original system.

###  IV-B Magnetic substrates for computation

A number of substrates based in ensembles of different types of magnetic
materials show interesting dynamic interplay between the elements arising from
various physical phenomena, and these substrates are promising candidates for
reservoir computing applications. There are a number of recent examples of
studies exploring the possibility of exploiting magnetic substrates for
computation [e.g., 29, 30]; however, we focus here on a specific paper by
Jensen et al. [5] concerned with exploring how to tune the dynamic behavior of
artificial spin ice (ASI) by varying the parameters of an external driving
field. For a recent review of the use of physical substrates for reservoir
computing, see Tanaka et al. [19].

ASI [31] consists of an array of coupled nanomagnets arranged on a two-
dimensional lattice. Each nanomagnet can be viewed as a single bit because of
its dipolar state. The state behavior of these nanomagnetic islands arises
from the small scale of their dimensions and their shape. Nanomagnets consist
of a single such domain, meaning the magnetization does not vary across the
magnet. Oftentimes, the nanomagnets will be fabricated with an elongated
rectangular shape, causing there to be two states that are energetically
favorable, termed spin up and spin down and corresponding to magnetization
along the longitudinal axis in the positive or negative direction. This shape
of nanomagnet was used in the square ASI array studied by Jensen et al. [5]
(Fig. 6).

Figure 6: Schematic showing the layout of the ASI studied by Jensen et al. [5]
(left). The number of unique states visited by the array was counted for
different external field parameters (right). Reproduced from [5].

Jensen et al. [5] perturbed this simulated nanomagnetic array with a time-
varying external magnetic field, $B(t)=A\sin\omega t$, at a direction
$45^{\circ}$ from the horizontal, and the field parameters $A$ and $\omega$
were tuned to achieve different types of dynamical behavior. Because each
nanomagnet in the system can take on one of two states, the overall state of
the system can be represented by 40 bits, yielding 240 possible states. To
quantify the complexity of the behavior of the system, 100 cycles of the
external field were applied to the system with different strengths $A$ and
frequencies $\omega$, and the number $S$ of unique states observed at the end
of every cycle was counted ($1\leq S\leq 100$). The results are shown in the
right-hand panel of Fig. 6.

For weak fields, none of the magnets switch their state ($S=1$), and for very
strong fields, all of the magnets switch at the half-cycle point and then
switch back ($S=1$ again). Additionally, at low frequencies, any transient
behavior has died out by the end of a single cycle, so the number of states
remains low, whereas very high frequencies produced chaotic behavior, with the
nanomagnets not having sufficient time to “keep up” with the switching of the
oscillating field direction. At the intermediate frequency of 100 MHz, long
transients were observed in a manner analogous to the complex CA behavior seen
in the right-hand panel of Fig. 3.

This type of substrate cannot show the same kind of long-range connectivity
that is observed in neuronal assemblies, as a single element of a magnetic
system can only directly affect other elements within a certain local radius,
whereas neurons can grow axons that can extend over very long distances within
the network. Thus, the “connectivity” of ASI is governed by quite a different
set of physical rules. However, the study described here demonstrates that
complex behaviors can be captured in this substrate and information can
propagate through the local interactions that occur between pairs of
nanomagnets. By modeling the essence of the behaviors of neural systems that
allow them to achieve the kinds of dynamics and optimal information processing
behaviors described in the previous section, we can drive the development of
magnetic substrates such as these towards a realm of greater efficiency and
power, with the possibility of capturing some of the information processing
capacity of the brain in engineerable computational substrates.

##  V Discussion

The fields of complexity and artificial life offer a great many tools to study
emergent behaviors and complex dynamics in different types of dynamical
systems. Close inspection of such systems—be they model systems like cellular
automata, natural systems like the brain, or fabricated systems like
artificial spin ice—reveals rich and varied behaviors that are challenging to
capture in simple metrics but beautiful to watch unfold.

With the use of models like graphs and CAs, we can capture the wide range of
possible dynamical behaviors observed in various physical systems and target
those behaviors that align with the hallmarks of computational power and edge-
of-chaos dynamics. This vanishingly small space of critical systems described
by Langton [2] and Wolfram [16] is accessible to us if we know how or where to
search, and tools like evolutionary algorithms put this possibility at our
fingertips. Furthermore, with the connection between neuronal avalanches and
criticality [25], we may also work backward from the product—the neuronal
system identified as complex—to the description of its connectivity and
response to inputs. From this understanding our focus may then turn to
eliciting from manmade substrates the same capacity for computation we see in
natural systems. However, it is important to remember that these models and
neuro-inspired substrates are not the brain, nor do they behave precisely as
the brain does—remember again Simon’s (1996) statement: “the artificial object
… turn[s] the same face to the outer system.” Interpretations of the behavior
we capture in our models must be tempered with this understanding: that as we
mimic, we do not precisely recreate, and there may be just as much to gain
from the differences as the similarities.

An artificial life approach can provide solutions to a wide range of practical
questions. The focus here has been on bio-inspired parallel and distributed
computation, with potential applications spanning from the implementation of
biologically plausible models for computation to the development of bio-
inspired computational substrates with the ability to learn and adapt,
offering a physical environment better suited for artificial intelligence
applications than conventional hardware. In addition, teaming up with
biologists and neuroscientists to better characterize the dynamics of the
brain may open the door to previously unconsidered diagnostic or clinical
tools.

But apart from the practical, these systems spark in many researchers an
innate and powerful curiosity needing no pragmatic outlet. We argue this drive
to understand how complex behaviors emerge from tauntingly simple components
and rulesets describing their interaction—coupled with a mind open to the
possibilities of what an exploration of these systems will reveal—is what will
ultimately prove fruitful in future research, giving us fodder for the
practical applications where such answers were not at first sought. There
still remains much for the tortoises to teach us.

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# Theoretical Aspects of the Quantum Neutrino

Stephen Parke∗ Chicago, USA
∗E-mail<EMAIL_ADDRESS>

###### Abstract

In this summary of my talk I will review the following the following three
theoretical aspects of the quantum neutrino: current status, why we need
precision measurements and neutrino oscillations amplitudes.

###### keywords:

Neutrino, Theory, Quantum

##  1 Current Status:

Circa 2017, it is now well established that neutrinos have mass and the that
the flavor or interactions states $\nu_{e},~{}\nu_{\mu}$ and $\nu_{\tau}$ are
mixtures of the the mass eigenstates or propagations states, unimaginatively
labelled $\nu_{1},~{}\nu_{2}$ and $\nu_{3}$. The interaction and propagation
states are related by an unitary matrix, the PMNS matrix, as follows:

$\displaystyle\left(\begin{array}[]{c}\nu_{e}\\\ \nu_{\mu}\\\
\nu_{\tau}\end{array}\right)$ $\displaystyle=$ $\displaystyle
U_{23}(\theta_{23},0)~{}U_{13}(\theta_{13},-\delta)~{}U_{12}(\theta_{12},0)\left(\begin{array}[]{c}\nu_{1}\\\
\nu_{2}\\\ \nu_{3}\end{array}\right).$ (7)

where the $U$’s are the usual complex rotation matrices given by
$[U_{mn}(\xi,\eta)]_{ij}=[1+(c_{\xi}-1)(\delta_{im}+\delta_{in})]\delta_{ij}+(s_{\xi}e^{i\eta})~{}\delta_{im}\delta_{jn}-(s_{\xi}e^{-i\eta})~{}\delta_{in}\delta_{jm}$.

Figure 1: The identity of the neutrino mass eigenstates or propagation states
is determined by their $\nu_{e}$ content: $\nu_{1}$ has the largest
($\sim$68%), $\nu_{2}$ is in the middle ($\sim$30%) and $\nu_{3}$ has the
least ($\sim$2%). The remaining components are a combination of $\nu_{\mu}$
and $\nu_{\tau}$ but it’s the $\nu_{e}$ fraction that defines the identity of
these states. Some, use the masses to label these states, but since we don’t
know both the mass orderings at this stage, using the electron flavor content
is simpler.

Placing the CP violating phase in the $U_{13}$ matrix is customary, however
oscillation physics is unaffected by placing this phase in $U_{23}$ or
$U_{12}$ since

$\displaystyle
U_{23}(\theta_{23},0)~{}U_{13}(\theta_{13},-\delta)~{}U_{12}(\theta_{12},0)$
$\displaystyle:=:$ $\displaystyle
U_{23}(\theta_{23},\delta)~{}U_{13}(\theta_{13},0)~{}U_{12}(\theta_{12},0)$
$\displaystyle:=:$ $\displaystyle
U_{23}(\theta_{23},0)~{}U_{13}(\theta_{13},0)~{}U_{12}(\theta_{12},\delta)$

where $:=:$ means equal after multiplying by a diagonal phase matrix on the
left and/or right hand side.

At the current time, it is most convenient to label the neutrino mass
eigenstates according to the size of $\nu_{e}$ fraction as is shown in Fig. 1.
With this choice for the mass eigenstates, it is natural to choose the order
of the $U_{\alpha j}$ matrices as in Eq, 7 so that the first row and third
column are simply, since these elements are most easily measured.

Since the neutrino oscillations with a $\Delta m^{2}\approx 2.5\times
10^{-3}{\rm eV}^{2}$ has only small amounts of $\nu_{e}$, it is $\nu_{1,2}$
that is has a $|\Delta m^{2}_{21}|\approx 7.5\times 10^{-5}{\rm eV}^{2}$. The
SNO experiment determined the mass ordering of $\nu_{1}$ and $\nu_{2}$,
$m_{2}>m_{1}$, see Fig. 2.

The remaining mass ordering, whether $m_{3}$ is larger or smaller than $m_{1}$
and $m_{2}$, is shown in Fig. 3. Current and future experiments such as
NO$\nu$A, JUNO, DUNE, T2HKK are designed to determine this mass ordering.

Figure 2: The solar mass ordering of $\nu_{1}$ and $\nu_{2}$ was determined by
the SNO experiment using the matter effects in the solar interior. The mass of
$\nu_{2}$ is larger than the mass of $\nu_{1}$ with $\Delta m^{2}_{21}\equiv
m^{2}_{2}-m^{2}_{1}\approx 7.5\times 10^{-5}{\rm eV}^{2}$. Here, I have
assumed that the $\nu_{\mu}$ fraction is equal to $\nu_{\tau}$ fraction for
both mass eigenstates.  Figure 3: The atmospheric mass ordering, sometimes
referred to as mass hierarchy, is the question of whether the mass of
$\nu_{3}$ is larger or smaller than $\nu_{2}$, $\nu_{1}$. $|\Delta
m^{2}_{31}|\approx|\Delta m^{2}_{32}|\approx 2.5\times 10^{-3}{\rm eV}^{2}$.
Here, I have assumed that the $\nu_{\mu}$ fraction is equal to $\nu_{\tau}$
fraction for all three mass eigenstates. Figure 4: Is the dominant flavor of
the mass eigenstate $\nu_{3}$, $\nu_{\mu}$ or $\nu_{\tau}$ ? If $\nu_{\tau}$
dominates then the parameter $\theta_{23}<\pi/4$, whereas if $\nu_{\mu}$
dominates then $\theta_{23}>\pi/4$. This is often referred to as the octant of
$\theta_{23}$ puzzle.

The octant of $\theta_{23}$, determines which flavor state dominates
$\nu_{3}$, see Fig. 4. Whereas the range of $\nu_{\mu}$ and $\nu_{\tau}$
components of $\nu_{1}$ and $\nu_{2}$ are determined not only by the allowed
range of $\theta_{23}$ but also by the CP violating phase $\delta$
($\cos\delta$ to be precise). There is significant depends on $\delta$ for the
magnitude of $U_{\mu 1}$, $U_{\mu 2}$, $U_{\tau 1}$ and $U_{\tau 2}$ elements
of the PMNS matrix. This fact is quite different than in the quark sector !

Figure 5: The $\nu_{\mu}$ and $\nu_{\tau}$ fractions of $\nu_{1}$ and
$\nu_{2}$ depends sensitively on both the value of $\theta_{23}$ and the CP
phase $\delta$. The flavor fractions vary the most for $\nu_{1}$, between
$\delta=0$, $\sin^{2}\theta_{23}=0.4$ (upper left) and $\delta=\pi$,
$\sin^{2}\theta_{23}=0.6$ (lower right). Whereas for $\nu_{2}$, the most
variation is between $\delta=\pi$, $\sin^{2}\theta_{23}=0.4$ (lower left) and
$\delta=0$, $\sin^{2}\theta_{23}=0.6$ (upper right). Factors of 2 to 3
differences are still allowed by the data. Note that is $\delta=\pm\pi/2$,
$\sin^{2}\theta_{23}=0.5$ (middle middle), then the $\nu_{\mu}$ and
$\nu_{\tau}$ content is the same for all three mass eigenstates.

In the quark sector only the magnitude of $U^{CKM}_{td}$ has any significant
depends on $\delta_{CKM}$. This occurs because of the hierarchy in the sizes
of the mixing angles in the CKM matrix: $\theta_{12}\sim\lambda$,
$\theta_{23}\sim\lambda^{2}$ and $\theta_{13}\sim\lambda^{3}$ where
$\lambda\approx 0.2$.

##  2 Why we need Precision Measurements

Four reasons for performing precision measurements:

###  2.1 For Discovery of New Physics

An experiment like ICECUBE can discover new physics in the flavor ratios of
their PeV neutrinos, if precise values of the predictions for the $\nu$SM are
known, see Fig. 6, from [1].

Figure 6: Variation of flavor ratios using our current uncertainty on $\delta$
(left panel) and $\theta_{ij}$, $\delta$ (right panel), from Bustamante,
Beacom and Winter.

###  2.2 Stress Test Three Neutrino paradigm

Compared to the Quark sector the unitarity of the PMNS matrix has only been
tested at the 10% level [2].

Figure 7: The current triangle (left) and normalization (right) unitarity
constraints on the elements of the PMNS matrix form Parke and Ross-Lonergan.
There is still plenty of room for new physics, such as a light sterile
neutrino.

###  2.3 Test Theoretical Neutrino Models

Fig. 8 shows how improvements on the measurements of the mixing angles and CP
violating phase can used to distinguish various models that could possibly
explain the mixings of the neutrinos.

Figure 8: Current status (left) and future status (right) of the labelled
models predictions for $\cos\delta$ from Girardi, Petcov and Titov [3].

###  2.4 Connection to Leptogenesis Understanding Universe

Fig. 9 gives the allowed region for a model [4] with the measured value of the
Baryon asymmetry of the universe on the neutrino parameters

Figure 9: Consistency of the model by Ballet, King, Pascoli, Prouse and Wang
which has the measured value of the Baryon asymmetry, through Leptogenesis,
with the measured values of the parameters $\theta_{13}$, $\Delta m^{2}_{21}$
and $\Delta m^{2}_{31}$.

##  3 Neutrino Amplitudes:

The neutrino oscillation probability can be written as

$\displaystyle P(\nu_{\alpha}\rightarrow\nu_{\beta})$ $\displaystyle=$
$\displaystyle|{\cal A}_{\alpha\beta}|^{2}\quad{\rm where}\quad{\cal
A}_{\alpha\beta}=\sum_{j}U^{*}_{\alpha j}U_{\beta j}e^{-im^{2}_{j}L/2E}$ (8)

For two flavors we simple obtain

$\displaystyle{\cal A}_{\alpha\alpha}$ $\displaystyle=$ $\displaystyle
1+2i~{}s^{2}_{\theta}~{}e^{+i\Delta}~{}\sin\Delta\quad{\rm and}\quad{\cal
A}_{\alpha\beta}=(2s_{\theta}c_{\theta})~{}\sin\Delta$

where $\Delta_{jk}=\Delta m^{2}_{jk}L/4E$.

For three flavors, there are an infinite number of ways of writing these
amplitudes which all give the same result. So we need an organizational
principle. It is known that the $\nu_{\tau}$ oscillation amplitudes can be
obtained from the $\nu_{\mu}$ oscillation amplitudes by making the following
replacements $s_{23}\leftrightarrow c_{23}$ and $\delta\rightarrow\delta+\pi$.
This follows from the form of the $U_{23}$ matrix on the left hand side of the
PMNS matrix, eqn. 7.

Similarly, if one considers the rotation matrix on the right hand side of the
PMNS matrix, $U_{12}$, there is a symmetry that leaves the amplitudes
invariant: the symmetry is $m^{2}_{1}\leftrightarrow m^{2}_{2}$,
$s_{12}\leftrightarrow c_{12}$ and $\delta\rightarrow\delta+\pi$ which follows
from

$\displaystyle U_{12}(\theta_{12},\delta)\left(\begin{array}[]{c}\nu_{1}\\\
\nu_{2}\end{array}\right)$ $\displaystyle=$ $\displaystyle
U_{12}(\pi/2+\theta_{12},\delta)\left(\begin{array}[]{c}-e^{i\delta}~{}\nu_{2}\\\
e^{-i\delta}~{}\nu_{1}\end{array}\right)=U_{12}(\pi/2-\theta_{12},\delta\pm\pi)\left(\begin{array}[]{c}e^{i\delta}~{}\nu_{2}\\\
-e^{-i\delta}~{}\nu_{1}\end{array}\right).$ (15)

To maintain this symmetry, use unitarity to remove the $U^{*}_{\alpha
3}U_{\beta 3}$ term in eqn. 8, giving

$\displaystyle{\cal
A}_{\alpha\beta}=\delta_{\alpha\beta}+(2i)\sum_{j=(1,2)}U^{*}_{\alpha
j}U_{\beta j}~{}e^{i\Delta_{3j}}\sin\Delta_{3j},$ (16)

then111Since the overall phase of an amplitude is arbitrary, there are
arbitrary choices for the overall phase of each amplitude.

$\displaystyle{\cal A}_{ee}$ $\displaystyle=$ $\displaystyle
1+(2i)~{}c^{2}_{13}~{}[~{}c^{2}_{12}~{}e^{i\Delta_{31}}\sin\Delta_{31}+s^{2}_{12}~{}e^{i\Delta_{32}}\sin\Delta_{32}~{}]$
$\displaystyle{\cal A}_{\mu\mu}$ $\displaystyle=$ $\displaystyle
1+(2i)~{}[~{}~{}(c^{2}_{23}c^{2}_{12}+s^{2}_{13}s^{2}_{12}s^{2}_{23})~{}e^{i\Delta_{32}}\sin\Delta_{32}$
(17) $\displaystyle\hskip
35.56593pt+~{}(c^{2}_{23}s^{2}_{12}+s^{2}_{13}c^{2}_{12}s^{2}_{23})~{}e^{i\Delta_{31}}\sin\Delta_{31}$
$\displaystyle\hskip
14.22636pt+~{}(s_{13}s_{12}c_{12}s_{23}c_{23}\cos\delta)~{}e^{i(\Delta_{31}+\Delta_{32})}\sin\Delta_{21}~{}]$
$\displaystyle{\cal A}_{\mu\tau}$ $\displaystyle=$
$\displaystyle(2c_{23}s_{23})~{}[~{}(s^{2}_{12}-s^{2}_{13}c^{2}_{12})e^{i\Delta_{31}}\sin\Delta_{31}+(c^{2}_{12}-s^{2}_{13}s^{2}_{12})e^{i\Delta_{32}}\sin\Delta_{32}~{}]$
$\displaystyle\hskip
14.22636pt-~{}(2s_{13}s_{12}c_{12})~{}[c^{2}_{23}e^{i\delta}-s^{2}_{23}e^{-i\delta}]~{}e^{i(\Delta_{31}+\Delta_{32})}~{}\sin\Delta_{21}$
$\displaystyle{\cal A}_{\mu e}$ $\displaystyle=$
$\displaystyle(2s_{23}s_{13}c_{13})~{}[~{}c^{2}_{12}e^{i\Delta_{31}}\sin\Delta_{31}+s^{2}_{12}e^{i\Delta_{32}}\sin\Delta_{32}~{}]$
$\displaystyle\hskip
28.45274pt+~{}(2c_{23}c_{13}s_{12}c_{12})~{}e^{i(\Delta_{31}+\Delta_{32}+\delta)}~{}\sin\Delta_{21}$

No approximation has been used to obtain these amplitudes and all of these
amplitudes explicitly satisfy the $1\leftrightarrow 2$ symmetry mentioned
earlier. Again as an organizing principle, I have separated only the terms
which involve $e^{\pm i\delta}$ as they always appear multiplied by
$\sin\Delta_{21}$( note
$\sin\Delta_{21}=e^{-i\Delta_{32}}\sin\Delta_{31}-e^{-i\Delta_{31}}\sin\Delta_{32}$).

If one uses the approximation that $\Delta_{32}\approx\Delta_{31}$ then we can
rewrite

$\displaystyle{\cal A}_{\mu e}$ $\displaystyle\approx$
$\displaystyle(2s_{23}s_{13}c_{13})~{}\sin\Delta_{31}+~{}(2c_{23}c_{13}s_{12}c_{12})~{}e^{i(\delta+\Delta_{32})}~{}\sin\Delta_{21}$

and then it’s simple to see that the CP violating term is given by

$\displaystyle\Delta
P_{~{}CP}=8~{}(s_{23}s_{13}c_{13})~{}(c_{23}c_{13}s_{12}c_{12})~{}\sin\delta~{}\sin\Delta_{21}~{}\sin\Delta_{31}~{}\sin\Delta_{32}$
(18)

In Fig. 10 we give a graphical representation of the amplitude for
$\nu_{\mu}\rightarrow\nu_{e}$ and the associated bi-probability plot.

Figure 10: The amplitude for $\nu_{\mu}\rightarrow\nu_{e}$ as well as
$\bar{\nu}_{\mu}\rightarrow\bar{\nu}_{e}$ and the associated bi-probability
plot.

##  4 Perturbative Approximation to the Neutrino Oscillation Probabilites in
Matter

In this section, a simple and accurate way to evaluate oscillation
probabilities, recently shown Denton, Minakata and Parke [5], is given.
Details as to the why’s and how’s of this method are contained in these
papers.

The mixing angles in matter, which we denote by a $\widetilde{\theta}_{13}$
and $\widetilde{\theta}_{12}$ here, can also be calculated in the following
way, using $\Delta m^{2}_{ee}\equiv\cos^{2}\theta_{12}\Delta
m^{2}_{31}+\sin^{2}\theta_{12}\Delta m^{2}_{32}$, as follows, see Addendum
[Parke:2018brr]:

$\displaystyle\cos 2\widetilde{\theta}_{13}$ $\displaystyle=$
$\displaystyle\frac{(\cos 2\theta_{13}-a/\Delta m^{2}_{ee})}{\sqrt{(\cos
2\theta_{13}-a/\Delta m^{2}_{ee})^{2}+\sin^{2}2\theta_{13}~{}}},$ (19)

where $~{}~{}a\equiv 2\sqrt{2}G_{F}N_{e}E_{\nu}~{}~{}$ is the standard matter
potential, and

$\displaystyle\cos 2\widetilde{\theta}_{12}$ $\displaystyle=$
$\displaystyle\frac{(\cos 2\theta_{12}-a^{\,\prime}/\Delta
m^{2}_{21})}{\sqrt{(\cos 2\theta_{12}-a^{\,\prime}/\Delta
m^{2}_{21})^{2}~{}+~{}\sin^{2}2\theta_{12}\cos^{2}(\widetilde{\theta}_{13}-\theta_{13})~{}~{}}},$
(20)

where $~{}~{}a^{\,\prime}\equiv a\,\cos^{2}\widetilde{\theta}_{13}+\Delta
m^{2}_{ee}\sin^{2}(\widetilde{\theta}_{13}-\theta_{13})~{}~{}$ is the
$\theta_{13}$-modified matter potential for the 1-2 sector. In these two
flavor rotations, both $\widetilde{\theta}_{13}$ and $\widetilde{\theta}_{12}$
are in range $[0,\pi/2]$.

$\theta_{23}$ and $\delta$ are unchanged in matter for this approximation.

From the neutrino mass squared eigenvalues in matter, given by

$\displaystyle\widetilde{m^{2}_{3}}$ $\displaystyle=$ $\displaystyle\Delta
m^{2}_{31}+(\,a-a^{\,\prime}\,),$ $\displaystyle\widetilde{m^{2}_{2}}$
$\displaystyle=$ $\displaystyle\frac{1}{2}(\Delta
m^{2}_{21}+\Delta\widetilde{m^{2}}_{21}+a^{\,\prime}\,),$ (21)
$\displaystyle\quad\widetilde{m^{2}_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\Delta
m^{2}_{21}-\Delta\widetilde{m^{2}}_{21}+a^{\,\prime}\,),$

it is simple to obtain the neutrino mass squared differences in matter, i.e.
the $\Delta m^{2}_{jk}$ in matter, which we denote by
$\Delta\,\widetilde{m^{2}}_{jk}$, which are given by

$\displaystyle\Delta\,\widetilde{m^{2}}_{21}$ $\displaystyle=$
$\displaystyle\Delta m^{2}_{21}\,\sqrt{(\cos 2\theta_{12}-a^{\,\prime}/\Delta
m^{2}_{21})^{2}~{}+~{}\sin^{2}2\theta_{12}\cos^{2}(\widetilde{\theta}_{13}-\theta_{13})~{}~{}},$
$\displaystyle\Delta\,\widetilde{m^{2}}_{31}$ $\displaystyle=$
$\displaystyle\Delta
m^{2}_{31}+(\,a-\frac{3}{2}a^{\,\prime}\,)+\frac{1}{2}\,\left(\,\Delta\widetilde{m^{2}}_{21}-\Delta
m^{2}_{21}~{}\right),$ (22) $\displaystyle\Delta\,\widetilde{m^{2}}_{32}$
$\displaystyle=$
$\displaystyle\Delta\,\widetilde{m^{2}}_{31}-\Delta\,\widetilde{m^{2}}_{21}.$

To see these expressions have the correct asymptotic forms, use the fact that
$(\Delta\,\widetilde{m^{2}}_{21}-\Delta m^{2}_{21})=|a^{\,\prime}|+{\cal
O}(\Delta m^{2}_{21})$, for $|a|\gg\Delta m^{2}_{21}$. Plots of the matter
mixing angles and mass squared differences are given in Fig. 11.

Figure 11: In the normal ordering (NO): Top left, the matter potentials, $a$
and $a^{\,\prime}$, top right, sine squared of mixing angles in matter,
$\sin^{2}\widetilde{\theta}_{jk}$, bottom left, the mass squared eigenvalues
in matter, $\widetilde{m^{2}}_{j}$, and bottom right, the mass squared
differences in matter, $\Delta\,\widetilde{m^{2}}_{jk}$. $E_{\nu}\geq
0~{}(E_{\nu}\leq 0)$ is for neutrinos (anti-neutrinos). $E_{\nu}=0$ is the
vacuum values for both neutrinos and anti-neutrinos.

To calculate the oscillation probabilities, to 0th order, use the above
$\Delta\,\widetilde{m^{2}}_{jk}$ instead of $\Delta m^{2}_{jk}$ and replace
the vacuum MNS matrix as follows

$\displaystyle U^{0}_{MNS}\equiv
U_{23}(\theta_{23})\,U_{13}(\theta_{13},\delta)\,U_{12}(\theta_{12})$
$\displaystyle\Rightarrow$ $\displaystyle U^{M}_{MNS}\equiv
U_{23}(\theta_{23})\,U_{13}(\
\widetilde{\theta}_{13},\delta)\,U_{12}(\widetilde{\theta}_{12}).$

That is, replace

$\displaystyle\Delta m^{2}_{jk}$ $\displaystyle\rightarrow$
$\displaystyle\Delta\,\widetilde{m^{2}}_{jk}$ $\displaystyle\theta_{13}$
$\displaystyle\rightarrow$ $\displaystyle\widetilde{\theta}_{13}$
$\displaystyle\theta_{12}$ $\displaystyle\rightarrow$
$\displaystyle\widetilde{\theta}_{12},$ (23)

$\theta_{23}$ and $\delta$ remain unchanged, it is that simple. We call this
the 0th order DMP approximation.

Figure 12: For normal ordering (NO), $\nu_{\mu}\rightarrow\nu_{e}$ appearance:
Top Left figure is for T2K , Top Right figure is NOvA, Bottom Left figure is
T2HKK, and Bottom Right is DUNE. In each figure, the top panel is exact
oscillation probability in matter, $P^{ex}_{mat}$ (blue dashes) from [6], the
zeroth order DMP approximation, $P^{\,0th}_{appx}$ (red dashes) from [5] and
the vacuum oscillation probability, $P_{vac}$ (black dots). The Middle panel
is difference between exact oscillation probabilities in matter and vacuum
(black dots), and the difference between exact and 0th DMP approximation
(solid red) and exact and 1st DMP approximation (solid magenta)
approximations. Bottom panel is similar to middle panel but plotting the
fractional differences, $\Delta P/\overline{P}$.

These expressions are valid for both NO, $\Delta m^{2}_{ee}>0$ and IO, $\Delta
m^{2}_{ee}<0$. For anti-neutrinos, just change the sign of $a$ and $\delta$.
Our expansion parameter is

$\displaystyle\left|\,\sin(\widetilde{\theta}_{13}-\theta_{13})~{}\sin\theta_{12}\cos\theta_{12}~{}\frac{\Delta
m^{2}_{21}}{\Delta m^{2}_{ee}}\,\right|\leq 0.015,$ (24)

which is small and vanishes in vacuum, so that our perturbation theory
reproduces the vacuum oscillation probabilities exactly.

If ${\cal A}_{\nu_{\alpha}\rightarrow\nu_{\beta}}(\Delta m^{2}_{31},\Delta
m^{2}_{21},\theta_{13},\theta_{12},\theta_{23},\delta)$ is the oscillation
amplitude in vacuum, see eq. 17, then ${\cal
A}_{\nu_{\alpha}\rightarrow\nu_{\beta}}(\Delta\,\widetilde{m^{2}}_{31},\Delta\,\widetilde{m^{2}}_{21},\widetilde{\theta}_{13},\widetilde{\theta}_{12},\theta_{23},\delta)$
is the oscillation probability in matter, i.e. use the same function but
replace the mass squared differences and mixing angles with the matter values
given in eq. 19 \- 22. The resulting oscillation probabilities are identical
to the zeroth order approximation given in Denton, Minakata and Parke [5].

In Fig. 12, I have given the exact and approximate oscillation probabilities
for the $\nu_{e}$ appearance channel for T2K[7] and T2HK[8], NOvA[9],
T2HKK[10] and DUNE[11].

##  5 Conclusions

To summarize:

  * •

from Nu1998 to now, tremendous experimental progress on Neutrino SM: more at
Nu2018 !

  * •

LSND Sterile Nu s neither confirmed or ruled out at acceptable CL: - ultra
short baseline reactor experiments.

  * •

Great Theoretical progress on understand many aspects of Quantum Neutrino
Physics: Oscillations, Decoherence, Oscillations Probabilities in Matter,
Leptogenesis.

  * •

Still searching for convincing model of Neutrino masses and mixings: with
testable and confirmed predictions !

##  6 Acknowledgements

I would like to thank the organizers, and especially Prof. Wang Wei, for the
wonderful hospitality so that I could attend this great conference.

This project has received funding/support from the European Union’s Horizon
2020 research and innovation programme under the Marie Sklodowska-Curie grant
agreement No 690575. This project has received funding/support from the
European Union’s Horizon 2020 research and innovation programme under the
Marie Sklodowska-Curie grant agreement No 674896.

Finally, I would like to thank my collaborator Prof. Xu Zhan from Tsinghua
University for a wonderful collaboration and friendship. We meet at the
International Symposium on Particle and Nuclear Physics, Beijing, China, Sep
2-7, 1985 and wrote a beautiful and significant paper [12] that was a
pioneering paper in the field of “Amplitudes”.

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