You should never underestimate how important it is to get service from a leading engineers and architects insurance company, whether your company is large or small. Given the current unsettled economic climate, engineers need to make every attempt to protect their personal finances as well as financial assets of the firm they work with. Making certain your business has all of its bases covered is a priority of any business owner. In the architecture area, this is vitally important as a small error or omission can cause massive amounts of expensive litigation. The price of said litigation is generally enough to wipe out a small company, even when you're found not liable in the end. The possible harm to your reputation is also something you need to think about. This means that engineers insurance is an essential ingredient to a secure future for your business. To assist you in navigating this difficult course, Fenner & Esler is an experienced engineers insurance company which will help you find an engineer insurance plan that will be exactly suited to your requirements.
The cookie cutter policies you'll find from other engineer insurance companies online prevent a lot of engineers from obtaining policies that are effective and are going to suit their needs. These one-size fits all policies are what cause many young structural engineer to make the career risking mistake of going without insurance completely. Sadly, many engineers just beginning make this mistake. As an alternative to paying for insurance services they won't need, they put away funds in the event of an emergency. However, there are many issues with this scenario. First of all, taking from the cash that was put away is much too easy, and should a serious issue arise unexpectedly, the financial security of the company is now in question. You can't ever be sure when you will require the coverage because lawsuits might be filed years after the completion of the project. Also, many people underestimate the amount of money they will have to spend in legal expenses, even when they're found not liable. Such lawsuits are known to drag on for years and can devastate or bankrupt your company, as well as the harm to your professional standing. Your best option is rather to seek the assistance of a trusted independent engineers insurance company like Fenner & Esler, since we are able to locate engineers insurance that will suit your exact requirements.
Finding the best engineer liability insurance company might be challenging when you take into consideration how many on-line insurance offers you will surely run into. Different insurance carriers have different exclusions, deductibles, and riders, which might make it difficult to compare the various policies they supply. If you need a complete policy for engineer insurance, it will usually be extremely costly and may over-insure engineers in small businesses or solo practices.
We are an independent engineer insurance broker and have years of experience working with design professionals. As a result, we realize that no architecture firms are alike. Their requirements for engineer liability insurance will vary as the nature of the projects they take and the experience of the professionals involved will result in different types of risks. At Fenner & Esler, we make an effort with every one of our clients to understand their business and listen to the things they're concerned about. Based upon this information, combined with our industry knowledge, we develop a comprehensive engineers errors and omissions insurance package with protection from A-rated carriers tailored to match the specific requirements of our clients. The benefit of an independent engineers insurance brokerage such as Fenner & Esler is that you will have the proper amount of insurance coverage at a price which you can afford.
With over 30 years experience providing engineers liability insurance, you'd be hard pressed to find a better independent engineer liability insurance company. We work in your favor as an independent engineer insurance company, because our first loyalty is to you and not an insurance provider. Partnering with an independent engineers liability insurance company like Fenner & Esler also supplies you with the ability to take advantage of our in depth knowledge in creating personalized engineers liability insurance policies that will also offer you flexibility throughout different stages your career.
With one-size-fits-all engineers insurance plans that you find from larger engineer insurance companies online, generally you don't get information on the design industry's best practices for risk management. If you decide to research this information yourself it is time consuming, and then you still must determine not only whether a specific recommendation pertains to your practice but whether it is consistent with your state's laws and ordinances. At Fenner & Esler, we will not just develop a personalized engineer errors and omissions insurance coverage policy to fit your needs but will also serve as your partner in risk management. We'll notify you of new industry practices that pertain to your practice, go over contracts, and offer regular guidance on ways you can protect your firm. This way, we can provide you with the peace of mind you need in order to focus on your business rather than being worried about frivolous litigation. When you hire us for engineers insurance, you can be certain you'll be working with an independent engineer insurance brokerage that you can trust with your practice.
We have strong relationships with solid A-rated engineer insurance coverage carriers, which is what makes us a great choice for an independent engineer insurance company. Our customers understand they can rely upon their engineer's professional liability insurance policy in the event it is ever required. For beginning engineers getting started in the business, we provide scalable professional liability insurance packages. As firms grow, we can adjust your coverage so that you are constantly protected in case a customer brings forth claims of negligence or faulty design. Later in your career, your policy could be altered to supply cost-effective protection in case a former client decides to file a lawsuit years following the completion of a project. This is one of the reasons we're a top choice for those looking for engineers insurance companies.
At Fenner and Esler, we don't ever take our solid reputation for granted. In addition to our services for engineers, we also supply environmental consultant insurance, surveyor insurance and engineers insurance, serving a wide range of professionals in the design industry. Whether you want to renew your engineers PL insurance, determine if you possess adequate engineers E&O insurance, or you want to investigate different options for a new engineers insurance policy, we will listen to your concerns so we recommend the best policy and risk management strategy. When you want to provide effective protection for your practice, contact Fenner & Esler, a top quality engineers insurance company you will discover.


POPSUGAR Reading Challenge Prompt: A book with a fruit or vegetable in the title.
Brief synopsis: Georgia Ford is a successful lawyer. One who came from a humble family who runs a successful small vineyard in Sonoma County. Weeks before her wedding's set to happen, she discovers a huge lie her fiancé has been keeping from her. This secret drives her back home to a place that's full of more secrets, and Georgia has to learn the hard way what's worth saving.
I honestly really liked Eight Hundred Grapes. I love a good story about family, and this was perfect chick lit. Laura Dave's writing is spot on, so I'm excited to read another book of hers I already own - Hello, Sunshine. The characters were relatable, even if I can't say I condoned all their choices. But really, what's a book like this without it's drama?
I enjoyed that every few chapters, it skipped back time a bit to give more insight about Georgia's father and how he became a winemaker, and how his choices shaped his life and the life of the vineyard.
The book was overall predictable, and a little corny, the way all the best chick lit and chick flicks are. And it definitely had me lusting for warmer weather. Eight Hundred Grapes is the perfect beach read, and I would highly recommend it for those of you looking for something to take on your upcoming travels.


STELLAR CORPSES ANNOUNCE NEW E.P. AND RELEASE DATE!
Stellar Corpses announced today that their brand new E.P. 'Hellbound Heart' will be released on Friday April 13th, 2018. Above you can see the album cover art that was done by: Jasmine Rain. It will be released through the newly formed Batcave Records which is run by Dusty of Stellar Corpses and Rene of The Brains.
Pre-order details are coming soon so be on the look out.


text:
In the mountainous regions of Central America, Clemson students get a glimpse of what it means to be health care providers.
By Hannah Halusker
Tackling Dementia
Faculty and students study the effectiveness of a reminiscence therapy program designed to address the debilitating effects of dementia.
By Michael Staton
A Tiger's Guide to Tailgating
A football Saturday in Clemson is like a party nowhere else. Here's a guide for making the most of your tailgating experience.
By Nancy Spitler
All/Fall 2018 Features/web exclusive
Looking for holiday gifts this year of online shopping? We've already done the heavy lifting for you by compiling a holiday gift guide chock full of products crafted by Clemson graduates — everything from custom skateboards, to cufflinks, to caviar. Happy gifting!
Miles of Heart
Saving Tigers
Something Very Fishy
Celebrating a Clemson historian's history
The team huddles, gathering strength for the battle ahead. They face no ordinary opponent today, but then again, they are no ordinary players. They are residents of Brookdale Senior Living Solutions; the opponent is their own failing memory.
A Transformative Place
In the midst of all the change, Clemson continues to be a transformative place for students.
Ben and Tommy's Excellent Frisbee Golf Course
In the '80s, Clemson's first unofficial Frisbee golf course began to take off, thanks to a couple of friends.
Cadence Count: Tiger Band by the Measure
The largest study of bullying prevention in U.S. schools reveals positive impact
Students receive prestigious Fulbright and Boren awards
'They gave me back my hope'
History in Plain Sight
Student researchers developing STEAM workshop model
Tinker Tailor Surgeon Spy: Fletcher Derrick '55
Far from Home: Corbey Dukes '84, M '94
Road Bots: Lionel Robert M '97
One for the Books: Megan Brown Clarke '04
On Ozark: Kevin L. Johnson '07
All-Star Administrator: Beth Goetz '96
My Clemson: Nina Kawar M '14
Q: Where do you live now, and what do you do? A: I'm…
When In ...
When in Washington, D.C.
Welcome to the nation's capital! My name is Charlotte Richardson,…
Reader-submitted Travel Stories
Tennessee: Candace Hall '10, Lyndsey Williams Mayweather '10, April Smith '08, Alexia Richardson '10 and Nyamekyer Sonja Badu '11
Candace Hall '10, Lyndsey Williams Mayweather '10, April Smith…
Clemson Travelers
Travelers Fall 2018
Catherine Myrick '91, Jenna Benson '19, Anne Miller Adams…
Wood Utilization + Design Institute adds Katerra as founding member
Sonoco Institute Announces New Partnership with Digimarc
CLEMSON WORLD PAST ISSUES
Summer 2020 Vol 74 Num 2
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Winter 2020 Vol 74 Num 1
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Fall 2019 Vol 73 Num 3
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Summer-Fall 2017 Vol 71 Num 3
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Spring 2017 Vol 71 Num 2
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Summer/Fall 2015 Vol 69 Num 3
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Spring 2015 Vol 69, Num 2
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Winter 2015 Vol 69, Num 1
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Summer/Fall 2014 Vol 68, Num 3
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Barker Commemorative Issue 2013
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Summer 2009 Vol 62, Num 3
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Fall 2008 Vol 61, Num 4
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https://i1.wp.com/clemson.world/wp-content/uploads/2005/01/2005_WINTER_CW.png?fit=685%2C891&ssl=1 891 685 CW Designer https://clemson.world/wp-content/uploads/2018/09/CW_Logo_NEW.png CW Designer2005-01-03 11:18:362017-09-26 16:16:54WINTER 2005 Vol 58, Num 1
https://i2.wp.com/clemson.world/wp-content/uploads/2004/01/2004_WINTER_CW.png?fit=583%2C759&ssl=1 759 583 CW Staff https://clemson.world/wp-content/uploads/2018/09/CW_Logo_NEW.png CW Staff2004-01-22 10:03:352017-09-26 16:24:01Winter 2004 Vol 57, Num 1
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redpajama_set_name: RedPajamaCommonCrawl


Browser Autofill feature lets you store user credentials like name, E-mail, Address, Phone number, etc. so that you don't have to fill it manually again and again. Although most users see Autofill feature as a convenience that provides ease while filling out credentials, it could also lead to big security risks leading to financial losses and a threat to personal data.
If you are aware of Chrome Autofill feature, you can observe that Chrome's autofill allows storing postal addresses (which contains information such as name, city, telephone, postal code, email address, etc. ) and credit card (which contains information such as cardholder name, number and expiration date).
Now, every piece of this data (except credit card information) can be synchronized with a Google account without you even noticing the same. This potentially increases the risk of your information being leaked to 3rd party attackers if your Google account credentials are leaked.
Imagine another scenario of a site requesting you to enter your name and email address to complete a form. While the autofill feature will automatically fill out the information against the name and email address, you will safely assume that this is the only information that you shared based on the fields that you saw on the form. But here is the where a potential leak may have happened.
The developer of a site can add hidden fields to a page, which is actually not "really hidden" from you but drawn outside the visible screen. Due to Autofill feature, your browser will automatically fill out fields which you can see and also the ones which cannot. So you always run the risk of getting your personal data shared without knowing though the hidden fields.
Below is a simple demonstration of form fields hidden from the user in Google Chrome. It shows that while only Name and Email were asked in the form, the Autofill feature submitted more information such as Email, Phone, Address and more once the user clicked on the "Submit" tab.
One of the unfeasible approaches would be to analyze the source code of the web page before submitting anything. However, this is not possible practically and also involves technical know-how. Hence, the best approach would be to disable autofill feature permanently.
Here is how you can disable autofill feature in Chrome.
While autofill feature is a tremendous time-saver, it is prone to data leaks and malware intrusions. It is highly recommended that you never let a browser take care of your passwords, no matter how safe they claim.
Though browsers let you interact with Internet World, they also open your system up to a variety of vulnerabilities and attacks. The browser can be an entry point for an attacker into your PC, hence securing your browser should always be a priority.
Here are eight tips to protect your browser.
For details visit Github. A bug report has also been filed with Mozilla here.


As you may recall, we added support for 3rd party OATH OTP clients to be used as an authentication mechanism with our service in the 16.2 release last month. The feature added in 16.2 allowed the Admin to enable a 3rd party client to be used by his/her end-users. However, this only worked for OTP clients for the user to manage on their own (e.g. add the client to the service by scanning a QR code from the User Portal). This works great for OTP clients (e.g. Google Authenticator) that support this model; however, not all OATH client (e.g. hard tokens) provide the end user with a way to add a QR code. In addition, many customers want to manage the OATH clients for their users.
With 16.3, Admins can now do a bulk upload of OATH tokens (under Settings > Authentication > Other > OATH Tokens) to register OATH clients with the service on behalf of the user. Admins can also remove these tokens later as needed.
With 16.3 we have added a plug-in that automates the configuration of the SharePoint Server to work with our service. In the past, admins would need to go through a bunch of manual steps to download and install a certificate from Centrify then manually configure their SharePoint Server. While this process was not difficult, it was error prone and time-consuming. Instead of spending 30 minutes to manually configure SharePoint, Admins can integrate their SharePoint server in about a minute by downloading the plug-in and executing a simple PowerShell script.


The Krupp last sold at Sotheby's on May 16th, 1968, for $305,000, to Elizabeth Taylor. The stone weighs 33.19 carats and is mounted in a ring. She wears it nearly every day and in every film since acquiring it; it was even animated into her special guest appearance on The Simpsons. Taylor is seen polishing her Academy Awards, we see her eye reflect, blinking, on the stone's facets.
Elizabeth Taylor wearing the Krupp ring.
Elizabeth Taylor also owned a 69.4-carat pear shaped, D-color, Interally Flawless diamond. It came up for auction at Sotheby's on October 23, 1969. At the sale, Cartier outbid Richard Burton who was bidding over the phone, but he later negotiated with them to the record breaking tune of $1,050,000 for the stone. When the stone changed hands, it was set in a ring. It was named the Taylor-Burton Diamond and mounted as a pendent on a $25,000 necklace by Harry Winston. There also was unprecedented publicity around this sale. In 1978, Elizabeth Taylor sold the stone, and the proceeds went to help build a hospital in Botswana, Africa. Sometime later that year Robert Mouawad purchased it.
Elizabeth Taylor flashes the Krupp Diamond to a group of reporters.
This is what it says in my volume of ART AT AUCTION The Year at Sotheby's and Parke-Barnett 1967-68: "A diamond ring, by Harry Winston. New York $305,000. From the collection of the late Vera Krupp von Bohlen und Halbach. The highest price ever paid [at the time] at auction for a diamond ring."


Marc joined MarinaSales Asset Advisors after several years as a principal at a long-standing mergers and acquisitions advisory firm. He has advised in the merger or sale of numerous businesses and properties throughout the United States. He manages with a thorough approach to client service and maintains exceptional standards for business excellence. He is a graduate of the University of Southern Maine.
Luke joined MarinaSales Asset Advisors with a background in financial analysis and market strategy. Prior to joining MSAA, he has spent past 10 years working in the financial services industry and consulting small businesses throughout southern Maine. He is a graduate of the University of Maine with a bachelors degree in Economics and Finance.


text:
\section{Summary}
This Appendix gives details of a number of items from the main text.
Section~\ref{sec:stg} gives detailed description of the crustacean STG with data shown in Fig.~1 of the main paper.
 Section~\ref{sec:chaos} classifies fast and slow chaos. Section~\ref{sec:prob} gives the
  mathematical detail of our probabilistic model. Section~\ref{sec:universality}
  gives a number of examples to illustrate  the universality
  of our results.
  Source code for the numerical simulations are available at \cite{bib:codes}.

\section{A more detailed description of the STG model}\label{sec:stg}

The stomatogastric ganglion (STG) of crustaceans is a well-studied
rhythmic motor circuit that controls movements of muscles involved in
chewing and filtering of food. It produces a remarkably stable output in
the face of changing internal and external conditions and has provided
deep insights into circuit and cellular level mechanisms involved in
achieving circuit robustness and homeostasis. The pyloric circuit within
the STG is a central pattern generator that produces a characteristic
triphasic pattern of activity. A pair of pyloric dilator (PD) neurons
bursts initiate the rhythm, and they are followed in phase by a lateral
pyloric (LP) neuron and finally by the pyloric neurons (PYs). All
neurons have large amplitude slow-wave membrane voltage oscillations and
fire a burst of action potentials at the peak of the depolarization
during normal activity (Fig. 1a of the main paper (MP)). There is cycle-to-cycle variability of
minor features of this pattern, but the order of firing, relative phases
and duty cycles (fraction of cycle period during which a neuron is
firing) are key to maintaining a functional output and are invariant.
Upon blocking of the neurohormonal inputs to the STG (decentralization),
this rhythmic bursting activity can change to an erratic spiking
behavior with sporadic bursts and a complete loss of rhythmicity. The
circuit is no longer functional in this state (Fig. 1b (MP)). From the
mathematical viewpoint, STG circuit activity shows hallmarks of chaos
both in the rhythmic and non-rhythmic conditions, as shown in the
transition probability analysis plots in the bottom row of Fig. 1 (MP).
Cycle-to-cycle variability of the spikes within a given burst,
especially in LP and PY spikes can be seen, which fill a region of space
in the three-variable plot even during normal triphasic activity.

\section{Classification of chaotic behaviors}\label{sec:chaos}

The main text concentrates on distinguishing between fast and slow chaos
that appear most meaningful for qualitative descriptions. Our analyses
led us to further identify the following types of chaotic behaviors:

{\em Weak chaos. } For a region of  $\alpha$ values
 starting around $2.7$ we observe a phenomenon we refer to as  {\em weak chaos} cf.
Fig.~2{a} (MP),
 ~3{a} (MP),
 in which the
 system is detected as chaotic using several measures of chaos, but
 does not noticeably alter the overall dynamics or functionality. The underlying mechanism
 for this chaos appears to be referred to as a weakly chaotic ring in~\cite{mira05},
 the same as the phenomenon found in~\cite{lorenz1989}.

 After $\alpha$ reaches the value $3$, the fast
 subsystem exhibits bistability, with a stable fixed point less than
 the resting membrane potential $\sigma$, and a chaotic attractor mostly
 located above $\sigma$. Again, the time trajectories are not significantly altered
 in terms of functionality.

   {\em Fast chaos.}
 Starting at around $\alpha>3.5$, the system shows
 relaxation oscillations through the strange attractor of the fast
 variable, cf. Figs.~2b (MP),
 3b (MP).
 However, the decay
 of the slow variable takes a relatively fixed time. Thus
 the interburst intervals
 appear periodic at the
 slow timescale, cf. Fig.~2b (MP), right,
 and the  chaos is restricted to fast timescales.

 {\em Slow chaos.}
 As $\alpha$ is increased towards 4, as in Fig.~2{c} (MP),
 the full system to escape
 the chaotic region through a new route, or {\em shortcut}, distinct from the
 classical relaxation cycle. The plot of the slow variable shows that
 it is unpredictable whether the
 escape from chaos will be through the classical or shortcut route.
 Slow chaos  persists for $\alpha>4$, as in Fig.~2{d} (MP).
 However, the frequency of the shortcut becomes greater as $\alpha$ increases.
 Since slow chaos results in unpredictable interburst intervals, slow chaos results
 in loss of functionality.
 Terman~\cite{terman92} described slow chaos and symbolic dynamics in a neural model
 with relaxation cycles
 as a system transitions from periodic bursting to spiking, but the mechanism is
 different and does not include the possibility of fast chaos.

{\em Hyperchaos.} When $\alpha$ is
 sufficiently away from the crisis, the frequency at which the system
 describes a full relaxation cycle vanishes, and the system fires
 repetitively without returning to the resting potential,
 cf. Figs.~2e (MP),
 3e (MP).
 The
 dynamics at $ \alpha$ above around $4.5$ displays {\em hyperchaos}, since it
 has two positive Lyapunov exponents, cf. Fig.
 2  (MP){ bottom};
 see~\cite{rossler79,stankevich20}.

\section{Probabilistic model}\label{sec:prob}
This section details the probabilistic model used to mimic the Rulkov map that has been proposed in the main paper.
We considered that the fast variable is a random uniform sampling of the support
of the fast attractor. Closed-form approximate expressions for this support
were used   in the case of the Rulkov map for our analysis. We provide
more detail on their derivation here.
\subsection{Geometric Analysis of the Fast Subsystem}
The slow-fast decomposition of the dynamics leads us to study the
dynamics of the fast variable (fast dynamics) when the slow variable $y$
is frozen. This one-dimensional dynamical system is given by the
function
\[x \mapsto F_{\alpha}(x,y)=\frac{\alpha}{1+x^2}+y.\]
The fixed points of
this equation are thus the roots of the cubic polynomial: are the
solutions of the cubic equation
\[x^3-x^2 y +x - y -\alpha=0,\]
and therefore, depending on $y$ and $\alpha$, the system has between 1
and 3 solutions, corresponding to intersections of  and fixed values of $y$ and
the curve
$y = x-\frac{\alpha}{1+x^2}$, forming the $x$-nullcline for the original map.
See the three branches Fig.~\ref{fig:graphicalnotation}, in solid red, dashed magenta, and solid yellow.
 The $x$-nullcline is also shown (in solid red) in Fig.~4{d,e} (MP) of the main text.

\begin{figure}
\includegraphics[width=.45\textwidth]{GraphicalNotations.pdf}
\caption{This figure illustrates the notation we are using in our probabilistic model. }
\label{fig:graphicalnotation}
\end{figure}

The fixed points located on the left branch of this cubic map are stable,
and those in the middle branch are always unstable,
with a positive multiplier larger than 1,
while the stability of the  right branch depends on $\alpha$ and
$y$~\footnote{This result is stated in the original Rulkov paper~\cite{rulkov2001regularization}.
	To prove the result it is helpful to define the function $	G (x) =F_{\alpha}(x,y) -x$; fixed points of $ F_\alpha(\cdot,y)$ correspond to zeros of $G$ and $\partial_x G =  \partial_x F -1$.
	Moreover,  the derivative of $G$ has the same sign as $ (1+x^2)^2\partial_x G =  -\left(1+x^2\right)^2-2 \alpha  x$, which is a  strictly concave function.
	In particular, if $ \alpha > \frac{8}{3 \sqrt{3}}$ then $G$ has two local extrema $ x_{min}  < -1 /\sqrt{3} < x_{max}<0$.

	As previously observed $G$ may have up to three zeros $\{x_l,x_c,x_r\}$, and  they entwined with the extrema: $ x_l \leq x_{min} \leq x_c \leq x_{max} \leq x_r$.

	Away from the bifurcations we have $ \partial_x G(x_c) >0$ and $ \partial_x G(x_l) <0$.
	Hence $ \partial_x F_\alpha(x_c,y) >1$, so $x_c$ is always an unstable fixed point.
	As $x_l <0$, then $ 0 < \partial_xF_\alpha(x_l,y)<1$, so $x_l$ is always a stable fixed point.
	The rightmost fixed point can be either stable or unstable; while it also satisfies $\partial_xF_\alpha(x_r,y)<1$ there is no lower bound on the derivative.
}. We denote by $u(y)$ the manifold of
unstable fixed points. The fast dynamics $x \mapsto F_{\alpha}(x,y)$ is
a unimodal map when considering $\alpha>0$, with a maximum reached at
$x=0$. We observe numerically that increasing $\alpha$ leads to a
period-doubling route to chaos, which is a classical result of a class
of unimodal maps as shown in~\cite{alligood-yorke}.

To compute a region enclosing the chaotic attractor $\mathcal{I}_\mu(y)=\{x: l_\mu(y)<x<r_\mu(y)\}$,
for each $y$,
we estimate an upper-bound using the maximal value of
$F_\alpha(\cdot,y)$. Thus the graph of $r_\mu(y)$ is given by
$\{(x_1,y_1)  = (F_\alpha(0,y),y-\mu)=(\alpha+y,y-\mu): y \in \mathbb{R} \}$.
The left side of the interval, meaning the
 graph of $l_\mu(y)$, is given by the image of the points $(x_1,y_1)$ on the graph of $r_\mu(y)$.
 That is,
the graph of $l_\mu$ is given by
\begin{eqnarray}
\{ &(&\l_\mu(y_2),y_2) = (F_\alpha(x_1,y_1),y_1-\mu(x_1+1)) \nonumber \\
& =&(F^2_\alpha(0,y),y_1-\mu(x_1+1)) \nonumber \\
&=& \left( \frac{\alpha}{1+(\alpha+y)^2}+y,y-\mu -\mu(\alpha+y +1) \right) :\nonumber \\
&y& \in \mathbb{R}  \} \, ,
\end{eqnarray}
cf. the solid blue and green lines in Fig~\ref{fig:graphicalnotation}, and
in the  left/right solid blue
lines in Fig.~4{d,e} (MP).

These curves superimposed with simulations are shown in Fig.~\ref{fig:prediction} and show an excellent agreement.

\begin{figure}
\includegraphics[width=.45\textwidth]{SupplementaryFigureFit.pdf}
\caption{The closed-form interval approximates closely the support of the strange attractor.
Same labels as in Fig. 3 (MP), with black lines representing $l_{\mu}(y)$ and $r_{\mu}(y)$. }
\label{fig:prediction}
\end{figure}

We know that the sequences of
iterates will be contained in this interval as long as
$F_\alpha(\mathcal{I}_\mu(y),y))\subset \mathcal{I}_\mu(y_1)$, which is in particular the case
when $l_\mu(y)$ is to the left of $u(y)$. Moreover, if
$x=0$ belongs to the support of a chaotic attractor, then this support
contains the boundaries of the interval $\mathcal{I}_\mu(y)$. In this
situation, an internal crisis will arise for parameters and a slow
variable $y$ such that $l_\mu(y)$ falls exactly on $u(y)$.
The two points at which this occurs are $Y_0$ and $Y_1$ depicted in Fig.~\ref{fig:graphicalnotation}.
In the
case $\mu = 0$, the condition yields
\begin{displaymath}
2 y^2 + 3\alpha y+ \alpha^2+2=0 \; .
\end{displaymath}
and the roots of the equation are given
$\frac{1}{4}(-3\alpha\pm \sqrt{\alpha^2-16})$,
with two real roots  when $\alpha > 4$ (and a single real root at $\alpha=4$).
Thus the chaotic attractor is contained
in an interval tangent to $u(y)$. Two real solutions correspond
to the two internal crises delineating the
region of occurrence of slow chaos within $y \in [Y_0,Y_1]$.
Note that if $\alpha > 4$ and $\mu > 0$, but sufficiently
small, then there will  still be  two ($\mu$-dependent) points $Y_0$ and $Y_1$
of intersection between $l_\mu(y)$ and $u(y)$. We will see below that in fact
for small fixed $\mu>0$, the interval $[Y_0,Y_1]$ persists for a small region of $\alpha<4$.
\subsection{A uniform random variable on the support}
Using the notation developed in the previous section,
for $y \in [Y_0,Y_1]$, if we model the system  such that it uniformly
samples the region $\mathcal{I}_\mu(y)$ as
it moves down along the slow flow, and consider the fraction of length of the interval $\mathcal{I}_\mu(y)$ to
the left of $u(y)$, then this provides an approximate probability for each the
iterate to lead to a shortcut trajectory. The probability of a long burst is thus
given by the probability to never fall to the left of $u(y)$.

Comparing the approximations to our empirical numerics, we see that for $\mu \ll 1$, there is  a
sharp boundary between the slow and
fast regimes, cf. Fig.~\ref{fig:scale_invariance} and Fig~4{a} (MP).
However, slow chaos arises in the fast system before
the slow system crisis at   $\alpha=4$, due to the fact that for $\mu>0$,
$\mathcal{I}_\mu(y)$ contains $\mathcal{I}_0(y)$.

Our probabilistic equation predicts the dependence of the probability of a
long burst as a function of $\alpha$ and $\mu$. In particular, we note that the
number of steps performed within the region $y \in [Y_0,Y_1]$ varies
inversely proportionally to $\mu$, which indicates an increase in the
probability of a short burst when the interval $ [Y_0,Y_1]$ does not vanish
with decreasing $\mu$. This is the case for $\alpha>4$, since that
region contains the set of values of slow variables for which $\mathcal{I}_0(y)$
is to the left of $u(y)$. In the same vein, for $\alpha< 4$,
decreasing $\mu$ enough will eventually prevent any intersection between
$\mathcal{I}_{\mu}(y)$ and $u(y)$ for all $y$, because of the continuous dependence
upon parameters and the fact that there is no intersection with
$\mathcal{I}_0(y)$.

We thus expect that, if $\mu$ is sufficiently close to 0, the probability of
a short burst will suddenly jump from 0 and 1 at $\alpha=4$. This is
visible in Fig.~4c (MP), where we show the joint dependence of
the probability of a short burst in $\alpha$ and $\mu$ (computed with numerical integration). Moreover, recall the formula of the probability of having a long burst,
$(1-p(\alpha,\mu))^{\Delta y(\alpha,\mu)/\mu}$ with $p(\alpha,\mu)$ the
fraction of the area of the region contained to the right of
the graph of $l_{\mu}(y)$ and to the left of the graph of $u(y)$,  where $\Delta
y(\alpha,\mu)$ is the size of the region of associated $y$ values.

Numerical simulations following this model predict a scale invariance of the probability of a small burst that
depends only on $(\alpha-4)/\mu$, which shows a clear agreement with simulations of the Rulkov model in Fig.~\ref{fig:scale_invariance}.

\begin{figure}
\includegraphics[height=0.6\columnwidth]{ScaleInvariance_Supplement.png}
\caption{Scale invariance property of the probability of a short burst. Data and
methods as in Fig.~4 (MP), but represented as a function of
$(\alpha-4)/\mu$, for $\mu=0.01$ and $\mu = 0.001$.  }
\label{fig:scale_invariance}
\end{figure}

\section{Network of Rulkov maps and Central Pattern Generators}
\begin{figure*}
\includegraphics[width=.9\textwidth]{Figure4Final}
\caption{
\label{fig:network}
Left: experimental data from~\cite{marder2}. Right: simulations of a
three neuron Rulkov model with inhibitory connectivity mimicking the
pyloric network (e). (a) Intact network, (b) Decentralized biological
network with irregular activity, (c) Rulkov model in the fast chaos regime
($\mu=0.01$, $\alpha=3.8$) and (d) Rulkov model in the fast chaos regime.
($\mu=0.01$, $\alpha=4$). (a2,b2,c2,d2): Voltage traces
for each neuron with, on the bottom, a stop-light representation of the spike
sequence. The associated transition matrices are shown in (a1,b1,c1,d1).
}
\end{figure*}
The neural network of the central pattern generators of the crab
discussed in the introduction involve neurons with vastly richer
dynamics than the Rulkov map, with heterogeneous properties and distinct
dynamics between neurons of different populations.  We however developed
a toy model of the crab STG made of three identical neurons described by the
Rulkov map, but with a biologically-inspired connectivity map (see
Fig.~\ref{fig:network}, right). In detail, the connectivity parameters were
chosen randomly as
\[\left(\begin{array}{ccc} 0 & -5.5 &-2.5\\-3.5&0 &
-5\\ -3&0&0 \end{array}\right)+U\]
where $U$ is a random matrix with
uniform distribution between $[-0.5,0.5]$ except on the diagonal.
While simplistic compared to the underlying biological phenomena, the Rulkov
system allows us to test the relevance of the transition to slow chaos in a
network without complications associated with intrinsic dynamics or
heterogeneities, although this would be a simple extension of our
analysis.

We compared our model in the regime of fast chaos and slow chaos to two
typical sequences recorded experimentally in the crab STG (data from
Kedia and Marder~\cite{marder2}) that qualitatively corresponded to
fast chaos (intact network Fig.~\ref{fig:network}(a)) or slow chaos
(decentralized network Fig.~\ref{fig:network}(b). We simulated our network and tested for the sequence of
neurons spiking, deemed to be a biologically relevant information. For
that purpose, we identified the onset of each burst and created a
sequence of burst initiations (colored circles below the neural dynamics
traces) identified using a threshold-crossing condition
(thresholds were adjusted according to the experimental traces for each
neuronal type, and fixed to $-1.5$ for the model), which led to the stop
lights dynamics represented below the traces. This sequence of spikes
was used to construct  a ``transition probability matrix'' $M$ that
contains as element $M_{ij}$ the empirical frequency that neuron $j$
spikes after neuron $i$. This is a stochastic matrix, since all elements
are non-negative and the sum over all columns on a given line is equal
to $1$. Given these matrices, we further computed an \emph{entropy}
level, computed as the average entropy of each line, or $-\frac {1}{3}
\sum_{i,j} M_{ij} \log(M_{ij})$ with the convention that $x\log(x)=0$
for $x=0$. A deterministic sequence gives an entropy equal to $0$, while
a maximal entropy of $\log(3)\approx 1.0986$ corresponds to a completely
random sequence of spikes. We compared our model in the regime of fast
chaos and slow chaos to two typical sequences that correspond to fast
chaos vs slow chaos recorded experimentally~\cite{marder2}. We observed
that both cases of fast chaos are certainly regular enough to ensure a
fully predictable sequence of spikes with probability 1 for the next
spike neural population to spike, and thus entropy zero. However, the
typical trace of a slow chaos experimental data show a very irregular
sequence, with evidence of the presence of shortcuts (shorts bursts)
very evocative of those observed in the Rulkov model (see left column,
bottom). The experimental dynamics are quite comparable to the
simplistic Rulkov model we designed (right column, bottom), and in
particular the transition matrix in this regime are non-trivial, with
strictly positive probabilities to have any neuron spiking after any
other neuron, and non-trivial entropy (0.67 for the data, 0.75 for the
model).

\section{Universality}\label{sec:universality}

This section consists of a series of examples with the fast-slow chaos to illustrate the universality of our
results. These examples are not restricted to maps, but also include differential equations models as well.

\subsection{Cubic map}
In~\cite{han17}, a one-dimensional, three-parameter cubic map was introduced and shown to feature complex dynamics. This system, given by the equation:
\[x_{n+1}=(-\beta x_n^3 +\alpha x_n+ \gamma)\]
 was shown to display chaos for specific values of the parameter $\gamma$ with or without internal crises depending on the value of $\alpha$. We thus exploited this dynamical richness to explore the existence of fast and slow chaos as well as transitions between those regimes. To this purpose, we extended the system to allow slow autonomous variation of $\gamma$, with linear dynamics according to $x$, in such a way that the system will tend to produce relaxation cycles through the strange attractors:
\[
\begin{cases}
x_{n+1}&=(-\beta x_n^3 +\alpha x_n+ \gamma_n)\phi(x_n)\\
\gamma_{n+1}&= \gamma_n+\varepsilon(-\kappa \gamma_n-x_n+\delta-\mu x_n^3).
\end{cases}
\]
In the equations above, we added for stability of the system a cubic term $-\mu x^3$ to the linear dynamics of $\gamma$ and a multiplicative term $\phi(x_n)$ to the $x$-dynamics preventing divergences when $x$ is large. Here, $\phi$ was taken a smooth relatively flat function with a plateau near $1$ for low absolute values of $x$ and near $0$ for larger values of $\vert x\vert$. Here, we chose
\[1-\frac 1 2 (\tanh(x-\theta)+\tanh(-x-\theta))\] with $\theta=2$.
Parameters were fixed to $\beta=0.5$, $\kappa=0.1$, $\delta=0.2$ and $\mu=0.9$
(These parameters were chosen arbitrarily to ensure creating appropriate relaxation cycles given the cubic dynamics).
The parameter $\varepsilon$ is the timescale ratio between $x$ and
$\gamma$, and was set to $0.01$ in our figure. For these parameters, the
fast variable typically displays chaos for appropriate values of
$\gamma$. It features hysteresis for $\alpha>1$, and hysteresis between
strange attractors starting from $\alpha=2$, associated with fast chaos
for the full system. A double crisis emerges at $\alpha=3$ with both
strange attractors hitting the unstable fixed point curve tangentially, and
leading to the onset of slow chaos (actually, for $\alpha$ slightly less
than this value when $\varepsilon>0$, as expected based on our analysis of the Rulkov map).

\begin{figure*}
\centerline{\includegraphics[width=.8\textwidth]{SupplementaryFigureCubicLabel}}
	\caption{Cubic map simulations for $\alpha\in\{2.5,2.8,3.2\}$, illustrating respectively fast chaos and a transition to slow chaos as the system approaches and crosses the crisis in the fast system (arising at $\alpha=3$). Top: phase space, with the fast attractor for various values of $\gamma$ (black) as well as the trajectory of the system for $\varepsilon=0.01$ (cyan). Bottom: evolution of the fast variable $x$ (blue) and slow variable $\gamma$ (orange) as a function of iterate number, illustrating clearly the emergence of fast chaos whereby macroscopically regular but microscopically chaotic behaviors emerge, and their transition to slow chaos with irregular dynamics when shortcuts in the relaxation cycles can arise during transient chaos near crises. }
	\label{fig:cubic}
\end{figure*}

\subsection{FitzHugh-Nagumo-Lorenz model}

To reproduce the scenario described in the paper in a continuous-time
dynamical system, we designed a model coupling a classical model of
neuronal relaxation cycle (the FitzHugh-Nagumo equation, variables
$(r,h)$) with the classical Lorenz equation (variables $(x,y,z)$) acting
as input to the fast dynamics:
\begin{equation}
\begin{cases}
	\dot{x} &= \sigma \,(y-x)\\
	\dot{y} &=-x\,z+\gamma \,r\, x-y\\
	\dot{z} &= x\,y-b\,z\\
	\\
	\dot{h}&=h-h^3/3-r+ x/\delta\\
	\dot{r}&=\varepsilon\,(h-r)
\end{cases}
\end{equation}
In this system, the variable $x$ in the Lorenz subsystem acts as an
input to the FitzHugh-Nagumo model, while the slow variable $r$ in the
FitzHugh-Nagumo model serves the role of the bifurcation parameter
(typically noted $\rho$) in the Lorenz system, therefore controlling the
chaotic or deterministic nature of the fast dynamics. The parameters
$\gamma$ and $\delta$ are scaling parameters that control the amplitude
of the coupling of those equations, with $\gamma$ fixed equal to $400$
allowing to generate our type of dynamics, an $\delta$ being varied. The parameter
$\varepsilon=0.01$ is the slow timescale ratio, and the Lorenz system
parameters are fixed to $\sigma=10$ and $b=2.66$ throughout our
simulations. By design, the Lorenz system will generate a fast chaotic
dynamics when the slow variable of the FitzHugh-Nagumo model $r$ is
large enough, which induces chaotic dynamics of the fast variable $h$.
The parameter $\delta$ controls the amplitude of the fast dynamics
strange attractor, and it can be chosen so as to induce an internal crisis.
For $\delta=30$, the chaotic attractors are significantly scaled down
and remain far from the unstable fixed point (Fig.~\ref{fig:lorenz}a2)
and the system generates relatively regular relaxation oscillations
through the strange attractors (fast chaos, Fig.~\ref{fig:lorenz}b2 and
c2). For $\delta=3$, the chaotic attractor is larger and in particular
contains two crises that open a shortcut avenue
(Fig.~\ref{fig:lorenz}a1), leading to erratic transitions between longer
and shorter relaxation cycles (slow chaos, Fig.~\ref{fig:lorenz}b1 and
c1). Simulations were performed in Matlab using a custom code based.
Numerical integration was done using the Euler scheme with timestep
0.001 and for a total duration of $4\,000$ time units after a transient of
$1\,000$ time units. Fast attractors were computed similarly over a range
of values of $r$ considered fixed, and starting with a range of initial conditions
in the vicinity of the fixed points of the fast system (fixed points were computed
using the Matlab vpasolve function and initial conditions were drawn randomly
according to a Gaussian centered the fixed point with standard deviation 0.01.

\begin{figure*}
\centerline{	\includegraphics[width=.8\textwidth]{Lorentz}}
	\caption{Simulations of the FitzHugh-Nagumo Lorenz model, for $\delta=3$ (top,
a1,b1,c1) or $\delta=30$ (bottom, a2,b2,c2). Left: fast attractors,
where each color represents a distinct orbit (a1,a2), Middle: projection of the five-dimensional
trajectory of the system in the space $(r,h,x)$ (b1,b2) and Right:
time-evolution of the FitzHugh-Nagumo variables $r$ and $h$ (c1,c2).  }
	\label{fig:lorenz}
\end{figure*}

\bibliographystyle{apsrev4-2}

\section*{Chaos and Homeostasis in nature}
Deterministic dynamical systems that feature chaotic dynamics are
notoriously characterized by irregular, aperiodic patterns of activity
and unpredictable behaviors highly sensitive to perturbations of parameters and initial conditions.
These erratic characteristics of chaotic dynamical systems
seem to be undesirable for physiological biological systems that
are required to maintain robust activity patterns to function. However,
the presence of chaotic dynamics has been reported in a number of
experiments on physiological behavior,
including key functions such as
cardiac contractions~\cite{hayashi1992chaotic} or brain
activity~\cite{korn}.  Some of these systems demonstrate highly
regular dynamics at longer timescales, with irregular fast activity, as
in example the stomatogastric ganglion (STG) of
crustaceans~\cite{marder2} described below. We explore here how mathematical chaotic
systems with multiple timescales may maintain enough macroscopic
regularity for function and how such systems transition to highly
erratic behaviors through a universal global bifurcation scenario.

Beyond its conceptual interest, this question has implications in
understanding possible consistency between chaos and \emph{homeostasis},
the process by which a variety of functions of living organisms are
regulated in response to changes in the
environment~\cite{bernard1879,may1998glutathione,dijk1999ageing,
aronoff2004glucose}. Various system-level feedback control mechanisms were
suggested to maintain macroscopic activity (see e.g., in neuroscience, ~\cite{turrigiano2004homeostatic,
cannon2016synaptic,marder2010coordinating,naude2013effects}). At
single-cell level and in the absence of exogenous mechanism,
homeostasis has been suggested to be opposed to chaos~\cite{goldberger1991normal},
and was associated with the existence of stable fixed points
that show little dependence to parameters~\cite{reed2017analysis,golubitsky2018homeostasis,
golubitsky2020infinitesimal} or robust limit cycles~\cite{thomas}. However, an
example of a chaotic system showing remarkable robustness is given by the
crustacean STG, a motor neural circuit producing a stable triphasic
bursting pattern of activity that controls movements of muscles involved
in chewing and filtering of food (see review~\cite{marder:07}). This
neural network is remarkably robust to changes in internal and external
conditions (Fig.~\ref{fig:data}a) and is able to maintain constant key
features for functional output such as the order of firing of the neural
populations, relative phases and duty cycles (see also the Appendix).
These macroscopic regularities arise despite a clear
cycle-to-cycle variability typical of chaotic systems (covering a dense
region of the phase space, Fig.~\ref{fig:data}a, bottom) which is also
reflected in various detailed models of the
system~\cite{alonso,alonso2}. However, this chaos can significantly
impact function when the neurohormonal inputs to the STG is blocked
(process called \emph{decentralization}), yielding an erratic spiking
behavior with sporadic bursts and a complete loss of rhythmicity,
associated with a loss of function (Fig.~\ref{fig:data}b).

\begin{figure}[h]
	\begin{center}
	\includegraphics[width=\textwidth]{Data.pdf}
	\end{center}
	\caption{Intracellular voltage traces from 3 pyloric circuit neurons of the
\emph{Cancer borealis} STG. (a) Chaotic behaviors in an intact system
showing rhythmic bursting, and (b) Erratic bursting/firing dynamics in a
decentralized network.}
	\label{fig:data}
\end{figure}

Mathematically, all formal definitions of chaos to date rely on fine
properties of the dynamical system~\cite{sander2015many}, regardless of
any possible regularity arising at slower timescales. We undertake here
the characterization of these different types of chaos in dynamical
systems, revealing the geometric structures underpinning chaotic activity which has
regular macroscopic behaviors, as well as clarifying universal transitions leading to
chaotic systems with erratic behavior at macroscopic scales.

\begin{figure}[t]
	\centering
	\includegraphics[width=.8\columnwidth]{Figure2label.pdf}
	\caption{Top: Chaotic trajectories of the Rulkov map, at parameters $\mu = 0.01,
	\sigma = -1$ with subfigures $a,b,c,d,e$ corresponding to
	$ \alpha = 2.698$,  $3.95$,  $4.00$,  $4.05$, $5.00$.
	 Left:~$x_n$; Right:~$y_n$;  Bottom: Lyapunov Exponents.}
	\label{fig:Lyap_Traces}
\end{figure}

For our investigations, we concentrate on the chaotic Rulkov
map~\cite{rulkov2001regularization}, a two-dimensional discrete
dynamical system reproducing key neuronal behaviors and preserving
mathematical tractability~\cite{ibarz2011map,courbage2010map}:
\begin{align}
x_{n+1} &= \frac{\alpha}{1+x_n^2} +y_n \\
y_{n+1} &= y_n - \mu \;  ( x_n - \sigma).
\end{align}
The dimensionless variable $x_n$ models the membrane potential of a neuron,
and $y_n$ is an adaptation variable accounting for slower gating
processes (the small parameter $\mu$ denotes the timescale ratio of
adaptation compared to voltage). The resting potential is noted $\sigma$
(hereafter, $\sigma=-1$). The parameter $\alpha$ controls the
excitability of the neuron and will serve as our main bifurcation
parameter in this study. As the neuron becomes more excitable, it
 transitions from a resting state to an invariant
cycle~\footnote{This cycle emerges through a Neimark-Sacker bifurcation
arising at $\alpha = -(1-\mu)(1+\sigma^2)^2/(2 \sigma)$ for almost every
$\mu<3$ (for our parameters, $\alpha$ slightly below 2} that rapidly
grows into a large amplitude relaxation cycle that eventually becomes
chaotic upon increase of excitability, as shown in
Fig.~\ref{fig:Lyap_Traces} (see~\cite[Section 2.1.2]{ibarz2011map}).
Similar to the STG case, two qualitatively significant types of chaotic
dynamics stand out: chaos that chiefly affects the fast
dynamics during a burst and regular slow behaviors (\emph{fast chaos}),
and chaos affecting the slow dynamics (\emph{slow chaos})
and that Lyapunov exponents fail to discriminate
(Fig.~\ref{fig:Lyap_Traces}, bottom).
Maslennikov and
Nekorkin were the first to report fast chaos
dynamics~\cite{maslennikov16}  in idealized mathematical models and in
discrete FitzHugh-Nagumo maps, while similar fast chaos dynamics in
forced systems were reported in a cubic map in~\cite{han17}.
Also note that Terman~\cite{terman92} described slow chaos and symbolic dynamics in a neural model
 with relaxation cycles
 as a system transitions from periodic bursting to spiking, but the mechanism is
 different and does not include the possibility of fast chaos.
To best present these dynamics, we
exploit the natural separation of timescales in the Rulkov map, and
separately consider the {\em fast dynamics} ($\mu=0$), corresponding to
collection of one-dimensional maps parameterized by the slow adaptation
variable $y$ that is seen as a parameter (red in
Fig.~\ref{fig:Attractor}). Depending on $y$, the fast dynamics
generically feature one stable equilibrium or three equilibria given by
the points with coordinates such that $y=x-\alpha/(1+x^2)$ (purple),
referred to as the $x$-nullcline. The $y$-nullcline $x=\sigma$ separates
regions of the phase plane where the slow dynamics increases or
decreases, governing the emergence of slow relaxation cycles.

We observe that fast chaos is directly related to the presence of
relaxation cycles through a strange attractor in the fast variable
(\ref{fig:Lyap_Traces}{b},\ref{fig:Attractor}{b}), as described in~\cite{maslennikov16}. These regimes
affect only the precise behavior of the voltage during a burst, with little
effect on the evolution of the slow variable, allowing the system to
maintain very close to periodic behavior at the slow timescale, cf.
Fig.~\ref{fig:Lyap_Traces}{b, right}. In contrast, as $\alpha$ is
increased towards 4, as in Fig.~\ref{fig:Lyap_Traces}{c}, the chaotic
attractor intersects with the unstable fixed point of the fast dynamics,
meaning that the system can terminate a burst before completing a full
relaxation cycle. Those {\em shortcuts} appear to be relatively
unpredictable and depend on the particular chaotic sequence arising at
the fast timescale, and they become more frequent as $\alpha$ is increased.

\begin{figure}[t]
	\centering
	\includegraphics[width=\columnwidth]{NewFigure2_2}
	\caption{Phase plane (blue) at $\mu=0.01$; nullclines (purple, yellow)
	and  attractor (red) for $\mu=0$. All have  $\sigma=-1$.
	Left to right,
	$\alpha = 2.698$, $3.95$, $4.00$, $4.05$, $5.00$. }
	\label{fig:Attractor}
\end{figure}

In the limit of timescale separation, the emergence of slow chaos
one would expect slow chaos to correspond to an internal crisis of the fast system;
we show in the
Appendix that it corresponds precisely at $\alpha=4$, where the fast
strange attractor hits the unstable fixed point of the fast dynamics.
For $\alpha>4$, we observe that the attractor is split into two regions
across a region where the fast dynamics converges to the lower stable
fixed point. However, in this non-chaotic region, transient
chaos arises along the ghost of the strange attractor, and
long transients emerge before the fast system converges
to the fixed point.
This slow convergence thus competes with the slow evolution of the
adaptation variable, allowing the full system to either converge to the
low-voltage fixed point or to go past the second internal crisis and
complete a full relaxation cycle. The system's progression  of full cycles and
shortcut forms a chaotic sequence, where the frequency at which the system
takes the shortcut varying as a function of $\alpha$.

\begin{figure*}
	\centering{
	\includegraphics[width=.8\linewidth]{Figure3Clean2.pdf}}
	\caption{Slow chaos and probabilistic model.
	a. Histogram of the inter-burst interval for $\mu=0.01$ and various values of $\alpha$. A bimodal distribution, corresponding with slow chaos, arises in a non-trivial range of $\alpha$ in the vicinity of the crisis point $\alpha=4$ (white box). b. One typical histogram for  $\mu=0.01$ and $\alpha=4$.  c. The shortcut probability simulated shows an excellent agreement with our analytical expression for the probabilistic model. d-e represent  the dynamical and probabilistic slow chaos scenario for $\alpha<4$ or $\alpha>4$, with d1/e1 showing the actual geometry of analytical curves derived (d1: $\alpha=3.9$, $\mu=0.05$, d2: $\alpha=4.1$ and $\mu=0.05$) and d2/e2 the topological structure.
	\label{fig:figure3}}
\end{figure*}
To further characterize the deterministic dynamics, let us consider the
sequence of interburst intervals along a given trajectory (see Fig.~\ref{fig:figure3}a).
A unimodal, non-Dirac distribution would correspond to fast chaotic trajectories, with distributions
centered at larger durations for fast chaos with no shortcut (cf. Fig.\ref{fig:figure3}a, small $\alpha$),
and centered at smaller durations for
fast chaos trajectories where the probability of shortcut is unity
(cf. Fig.\ref{fig:figure3}a, small $\alpha$).
In contrast, slow chaos corresponds to
bimodal distributions of interburst intervals, corresponding to trajectories composed of both
long bursts and shortcuts (white box in Fig.~\ref{fig:figure3}a and in~\ref{fig:figure3}b).

Based on these observations, we designed a probabilistic model to
predict the emergence of slow chaos and estimate the shortcut
probability. Our stochastic model approximates that the fast chaotic dynamics
as a random Bernoulli sequence (independent random variables) uniformly sampling
the interval of possible values for the fast variable along a relaxation cycle:
\[\begin{cases}
 x_{n+1}=\begin{cases}
 	\mathbb{U}_{\mathcal{I}_\mu(y_n)} & x_n>u(y_n)\\
	F_\alpha(x_n,y_n) & x_n\leq u(y_n)\\
	\end{cases}\\
 y_{n+1}=y_n - \mu(x_n-\sigma)
\end{cases}\]
where  $F_\alpha(x,y)= \frac{\alpha}{1+x^2}+y$ is the nonlinear fast dynamics, $u(y)$ is the unstable
fixed point of the fast dynamics, and $\mathbb{U}_I$ is a uniform random variable on the interval
$\mathcal{I}_\mu(y)$ of possible values taken by the fast variable during the high voltage portion of the burst.
Noting that the fast variable takes its maximum at $x=0$, this envelope is approximated by the first and
second iterates of $x=0$, yielding a closed-form expression that closely matches the numerically
observed support of the fast variable (see Appendix). The
significance of this representation is that it provides
a quantification of the relative occurrence of a shortcut between the two crises, where the chaotic system
provides no information.

In detail, if the attractor never intersects the unstable fixed point (i.e.,
$l_\mu(y)>u(y)$ for all $y$ where both functions are defined), long bursts
arise systematically, and the system displays fast chaos. However, if there exists an interval
of values $y \in [Y_0,Y_1]$ such that $l_\mu(y)<u(y)$, an iterate of the system
leads to a shortcut with probability $\rho_\mu(y)=(u(y)-l_\mu(y))/(r_\mu(y)-l_\mu(y))$.
The probability of not taking the shortcut will thus be given by the
product of $\prod_{n=n_0}^{n_1} (1-\rho_\mu(y_n))$ where $n_0$ is the time
$y_n$ enters $[Y_0,Y_1]$, and $n_1$ is the exit time. While this can be
computed explicitly, we further simplify the problem by considering an
averaged shortcut probability computed as fraction of surface area $p$
to the right of the unstable fixed point (green region in Figure
\ref{fig:figure3}d1\&e1), and the number of iterates is estimated as
$N=(Y_1-Y_0)/\mu$, the probability of a long burst is $P_l=(1-p)^{N}$.
Numerical calculation of $p$ under this assumption shows very good
agreement with the original simulations of the  Rulkov model
(cf. the Appendix, where we detail the exact scaling). Going
further, we observe that the exponential dependence in $1/\mu$ makes the
probability switch from $0$ to $1$ very rapidly upon variation of the
time scale parameter $\mu$ at a given value of $\alpha$ (i.e., for $p$
fixed). This indicates that in the limit of timescale separation, the
system always displays fast chaos, with long bursts for $\alpha<4$
and short bursts  for $\alpha>4$. As $\mu$ is decreased, slow chaos occurs
on a smaller interval of $\alpha$, as
shown in Figure \ref{fig:figure3}c. Disentangling the dependence of $p$,
$Y_0$ and $Y_1$ in $\alpha$ is a complex task and likely depends on the
particular system studied. In the Rulkov model, our numerical simulations
suggest that, for small $\mu$, the size of the  interval of
values of $\alpha$ associated with slow chaos is proportional to $\mu$, and
the profile of the probability of taking of a shortcut is scale
invariant when properly rescaling the nonlinearity $\alpha$. (See Appendix.)

This approach provides a very accurate value, for $\mu>0$, of the emergence of
slow chaotic trajectories for typical $\alpha<4$. For any fixed value of $\alpha$, there
exists $\mu$ sufficiently small for which the system displays fast chaos only,
characterized by relaxation oscillations. Considering a coarse-grained representation of the dynamics \cite{gedeon}, relaxation cycles in this regime are the concatenation of 4 dynamical sections (see Fig.~\ref{fig:figure3}d1,2):
B1 (slow rise of the adaptation variable preceding a burst), T1 (transition from quiescence to burst),
B2 (bursting behavior with decay of the slow variable) and T2 (transition from bursting to quiescence).
However, for larger values of $\mu$ and $\alpha<4$ sufficiently close to $4$, we find that \
$\mathcal{I}_{\mu}(y)$ intersects the unstable fixed point and allows for an additional, yet unpredictable
transition between branches B1 and B2 (labeled T3) appears, allowing the yielding the emergence of
short bursts of type B1-T1-B2-T3. For $\alpha>4$, shortcuts exist for arbitrarily low values
of $\mu$ (and actually become the most likely outcome as $\mu\to 0$, and coarse-grained transition graph is slightly modified (Fig~\ref{fig:figure3}e). In that case, the transition
region from bursting to quiescence is split into two disconnected
pieces B2 and B3, with returns to quiescence denoted by T2 and T3. For $\mu$
sufficiently small, fast chaos occurs with relaxation cycle B2-T3-B1-T1,
but when $\mu>0$, the transient chaotic behavior may prevent the
system from undergoing transition T3; in this case, cycles transition to B3 through
a new path B4, yielding a long bursting relaxation cycle
B2-B4-B3-T2-B1-T1.

The fine analysis that was made possible by the simplicity of the Rulkov
map reveals a universal phenomenon: relaxation cycles through fast
strange attractors will yield mathematically chaotic behaviors that
appear regular at larger timescales, and these phenomena will breakdown
at internal crises of strange attractors where chaotic alternations of
longer and shorter cycles will emerge. To provide further evidence for this universality,
we have shown that this behavior occurs in two other models: a cubic discrete dynamical
showing a double crisis; and  a 5-dimensional FitzHugh-Nagumo neuron model
coupled with the Lorentz system at a fast timescale.
These models are described in detail in the Appendix, with simulations show in
Appendix Figure 5.

Overall in this paper, we have observed that while chaos is usually associated with
disorder, the relationship of macroscopic patterns and chaos can be more
complex,  and in particular mathematically chaotic systems can display
regular and robust dynamics at slow timescales. We identified the
external crises of strange attractor global bifurcation governed sharp
transitions between such relatively regular chaotic dynamics and erratic
behaviors at slow timescales. In these regimes, erratic alternations
between long and short bursts are governed by long irregular transients
around the ghost of a strange attractor, a scenario that we sharply
validated with the introduction of a purely probabilistic model from
which we predicted the frequency of occurrence of shorter bursts. Both
slow and fast chaos are universal phenomena that arise in a variety of
maps and continuous systems. This phenomenon adds up to a wide
literature on chaotic dynamics of neural systems, that finely
characterized micro- and macro-chaotic structures in bifurcation
diagrams~\cite{barrio14} and the fine dynamics of chaotic alternations
in bursts at spike-adding transitions in slow-fast systems of
bursters~\cite{barrio21,medvedev,serrano21}, with a possible key role of
canards~\cite{shilnikov03}. Remarkable works also finely investigated
the complex chaotic dynamics of models of pancreatic beta-cells
systems~\cite{duarte09}, and argued that another global bifurcation
called interior crises of strange attractors could be related to some
sudden changes in bursting behaviors~\cite{duarte17}.
We showed
here that transitions between fast- and slow-chaos were universal and
could be found in maps as well as ordinary differential equations, and
yielded dynamics very similar to what is observed experimentally in the
crab nervous system, which we further illustrate in the Appendix with a
3-neuron model and various quantifications of data and numerical
simulations. This work motivates further analysis of the dynamics of
neural networks at chaotic crises, fine analyses of realistic neuron
models and data to assess compatibility with the present bifurcation
scenario, analysis of the robustness of bursts in various regimes, and
fine mathematical comparisons between the different mechanisms of
chaotic bursting generation.

\section*{Acknowledgements}
JJ was supported in part by NIH T32 NS007292.
ES was partially supported by the Simons Foundation under Award 636383. JT was partially supported by NSF DMS 1951369.
JJ and ES acknowledge support from NSF grant DMS-1440140 while they were at
residence at the Mathematical Sciences Research Institute in Berkeley, CA, during
the Fall 2018 semester. S.K. was supported by funding from the National Institutes of Health (R35NS 097343).

\section*{Appendix}
\input Appendix

\bibliographystyle{apsrev4-2}

redpajama_set_name: RedPajamaArXiv


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Home - Uncategorized - Tribeca Film Festival documentary debut features BrainGate2 neurotech subject
Tribeca Film Festival documentary debut features BrainGate2 neurotech subject
Research is consortium of Case Western Reserve University, the Cleveland Functional Electrical Stimulation Center at the Louis Stokes Cleveland VA Medical Center and University Hospitals Cleveland Medical Center
CLEVELAND—MAY 2, 2019—I AM HUMAN, a documentary debuting today at the 2019 Tribeca Film Festival in New York, examines the realities of neurotechnology, chronicling three research participants with experimental implantable brain treatment. (https://youtu.be/8SZbSimAHs0)
One of the participants was Bill Kochevar, who, two years ago, was the focal point of published BrainGate2 research led by Case Western Reserve University, the Cleveland Functional Electrical Stimulation (FES) Center at the Louis Stokes Cleveland VA Medical Center and University Hospitals Cleveland Medical Center (UH).
Kochevar, a U.S. Navy veteran who was paralyzed below his shoulders in a bicycling accident, worked with investigators regularly down the hall from his home at the Cleveland VA Medical Center in the Spinal Cord Injury Long Term Care Unit.
He is believed to have been the first person with quadriplegia in the world to have arm and hand movements restored with the help of two temporarily implanted technologies: A brain-computer interface with recording electrodes under his skull, and a functional electrical stimulation (FES) system* activating his arm and hand, reconnect his brain to paralyzed muscles.
Bob Kirsch, the Allen H. and Constance T. Ford Professor, chair of Case Western Reserve's Department of Biomedical Engineering and executive director of the FES Center, and Bolu Ajiboye, assistant professor of Biomedical Engineering, are principal investigators of the ongoing research.
Dustin Tyler, the Kent H. Smith Professor of Biomedical Engineering, was also interviewed for the film and appears as a "subject matter expert" commenting on the field of neurotech research.
The documentary, directed and produced by Taryn Southern and Elena Gaby, examines the promises, as well as the challenges, of neurotechnology, according to the production's press materials. Screenings are scheduled today through May 4 during the Tribeca Film Festival. A Cleveland premiere is being discussed for later this year.
"Advancements in neurotechnology are revolutionizing what it means to be human," according to the festival's description of the film. "Following three subjects who undergo brain interface (research to restore lost function), I AM HUMAN examines the ethical quandaries in brain exploration and cognitive evolution."
Along with Case Western Reserve, the Louis Stokes Cleveland VA Medical Center and the Cleveland FES Center, the film features institutions including Duke University, the University of Washington and The Wyss Center in Geneva, Switzerland.
After the premier screening, Ajboye will participate in a discussion moderated by Alex Klokus, chief executive officer of Futurism, with fellow panelists Southern, Gaby, neurotech entrepreneur Bryan Johnson, Toronto Western Hospital neurosurgeon Andres Lozano and Duke University Professor of Law and Philosophy Nita Farahany.
The other two participants featured in the film are a former Canadian government information technology officer, who undergoes a novel bionic eye implantation to regain sight, and a former artist with degenerative Parkinson's Disease, who considers deep-brain stimulation as a solution to her worsening symptoms.
*CAUTION: Investigational Device. Limited by Federal Law to Investigational Use.
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The NHS today withdrew all walking aids after a study found that 59% of people aged over 65 died within 10 years of starting to use them. Doctors have recalled all walking sticks currently in use in the hope of prolonging the lives of thousands of elderly, barely-mobile patients, but admit that for some the damage may already have been done. They also predict that recovering the items could take a while as many patients live within walking distance of their local hospital.
Help the Aged has been quick to attack the Government for what it has labelled a 'knee-jerk' response, but a planned protest march has had to be cancelled due to a lack of support.


text:
The (Virtue) Ethics of Private Property: A Framework & Implications
NEW PERSPECTIVES ON PROPERTY LAW, OBLIGATIONS AND RESTITUTION, A. Hudson, ed., London, Cavendish Press, pp. 39-67, 2003
15 Pages Posted: 11 Feb 2011 Last revised: 12 Jun 2012
This paper is part of a larger attempt to retrieve elements of the justification for private property that too frequently have been played down in legal and political discourse. These elements of justificatory argument are crucial to an adequate understanding of the institution of private property, its various practical manifestations, as well as its evolution and role in contemporary Western society. The argument here defended is that the concept of private property has an ethical dimension that is inherent in its very essence: in lay terms, private property is as much about duties, obligations and ends as it is about rights. This paper will sketch some of the theoretical underpinnings of such a re-constructive project, as well as set out some of the implications of the analysis at both conceptual and justificatory level. Related texts outline some of these elements and demonstrate how they are essential to the theoretical underpinnings and practical modalities of property rights and obligations, and set out definitional amendments consistent with the larger argument.
As a group, these studies will attempt to restore this group of non-rights-based aspects to their proper position alongside the rights-based elements of this institution, arguing that the former contributes to a larger functional moralism in the institution of private property, as these elements are directly related to societal goals and values. The project is in part a study of the theory of and justifications for private property, and in part an analysis of how the justifications and explanations for private property should inform, and be informed by, legal discourse. It is a self-consciously revisionist argument that seeks to re-tell the theoretical and practical "story" of private property. As the story is retold, it will be clear that there are elements of private property that are clustered around certain standards of conduct inhering in the concept and oriented by the general goals or virtues governing the way property ought to be used. These include not only the requirements that a community can place on specific property owners for the good of the community and the justifications for doing so, but also those requirements that emanate from the nature of the property relationship and the nature and characteristics of the particular resource. In addition, these normative claims bear not only on the use of resources but, even more directly, on their allocation.
Keywords: private property, virtue ethics, rights-based ethics, duty-based ethics, social obligations
Lametti, David, The (Virtue) Ethics of Private Property: A Framework & Implications (2003). NEW PERSPECTIVES ON PROPERTY LAW, OBLIGATIONS AND RESTITUTION, A. Hudson, ed., London, Cavendish Press, pp. 39-67, 2003, Available at SSRN: https://ssrn.com/abstract=1758893


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My iPod Mini looks like this one.
I bought my iPod Mini from the Apple Store at Arden Mall in Sacramento last September. I filled it with my favorite tunes, yes, the ones I grew up with. The 4GB harddisk is okay with me since I don't find the need to tote along all my music. It's great especially during my daily two hour commute to work as it blocks off all the undesirable noise in public transport like gossiping fellow passengers, Pinoy DJs trying to sound like Americans; I'm sure you get the picture!
I also bought a Sennheisser MX450 to even enhance my listening experience. It improved the bass sound.
I also used the iTunes AAC format to compress my music. There's a little quality loss but I can live with it. Applelossless is very good but it eats up a lot of HD space. That makes MP3 still the best format to compress your music to make them portable. Of course had I bought the full sized iPod with a 20 or 40 GB harddisk, space won't be a problem at all.
One of the things I love about my iPodMini is it's so easy to use! The click-wheel is a work of genius. You can smoothly change music by the flick of your thumb. What more, you can create an instant playlist through the On-The-Go menu. There is no way you can get bored with your music because you can surprise yourself by turning on the Shuffle mode.
I only have one little neg comment about this thingamajig --- battery life! The blurb says battery life is good for 8 hours for one charging cycle. The longest I've had is a mere 5 hours. And I need 3 hours to get a full charge!!! Beware, the battery is easily used up if you keep on pushing the forward button. The website explained that it has something to do with the cache memory; the iPod saves the next song in its cache and if you forward the song, the cache needs to store another song. This process eats up a lot of the battery's energy. So to avoid this from happening, I use the On-the-Go playlist a lot.


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Business / Money / News / security / technology / Uncategorized
Do Not Track Is Still Tracking
by Areeb Soo Yasir · Published October 17, 2018 · Updated October 18, 2018
Ever click the "Do Not Track' function in your browser settings assuming that it's stopping pages and websites from 'tracking you and your online activity'? Any reasonable person would think this. Except it's still tracking you and the sites that you are using know, you turned it on. In fact, they're not only tracking you, they're tracking the fact that you're trying to protect your privacy.
There should be standard protocol set when it comes to your privacy and since the "internet of things" is somewhat new and trendy, people don't get that it's actually a huge security nightmare- for you.
A survey by Forrester Research reports that "A quarter of American adults use the "Do Not Track" function to protect their privacy"- thats a huge number of people trying to stay safe and secure, and yet it's done nothing for them. By normal standards, this is pretty bad. Providing a service/function thats falsely providing you with nothing.
Every site you visit, you are being watched. The 'incognito mode' or 'private mode' is a prime example of this. It's not incognito or private at all- in fact your ISP and apps that you've downloaded and given permission to are all watching and collecting that data. They know that "John Smith" went on that porn site, watched this much stuff with these keywords for this long- and that data gets sold to parties that have an interest in that data. You think you're names not attached to it? It is.
These 'functions' are designed to generate the feelings of 'security and privacy' when it's actually watching you still. It's like telling you that you're inside a safe metal bunker but it's actually just cardboard spray painted to look like metal.
The intention started off nice. It was supposed to be the "Do Not Call" list of the internet and it was created by consumer advocates. But alas very few sites honor the "Do Not Track" function. Pinterest and Medium for example are two companies that do respect it- Medium won't send the data to 3rd parties which is a pretty big deal.
Others likes Twitter, Yahoo and the infinite sites of the internet do not honor it. Google, Facebook, Pornhub and a ton of other tank sites wouldn't even consider it. Google tracks in the name of 'better searches' even though Chrome has an option to turn off tracking, they still track you…. Facebook as we know has had their issues with data leaks, breaches and flat out privacy butchering – some of which you agree to in their TOS.
You have to ask yourself a serious question: Is Google, Facebook and all these free services just pumping in millions of dollars in infrastructure and people to create something for you to use for free? There is a price you pay, and that is your data, your privacy, your security.
This article gives you a really good in depth look into the 'Do Not Track' function and the details of why nearly all sites ignore it.
"Why do we have this meaningless option in browsers? The main reason why Do Not Track, or DNT, as insiders call it, became a useless tool is that the government refused to step in and give it any kind of legal authority. If a telemarketer violates the Do Not Call list, they can be fined up to $16,000 per violation. There is no penalty for ignoring Do Not Track.
From Gizmodo
Percentage of visitors to Gizmodo Media Group's sites whose browsers are requesting they not be tracked
In 2010, the Federal Trade Commission endorsed the idea of Do Not Track, but rather than mandating its creation, the Obama administration encouraged industry to figure out how it should work via a "multistakeholder process" that was overseen by W3C, an international non-governmental organization that develops technical standards for the web. It wound up being an absolutely terrible idea."
""Do Not Track could have succeeded only if there had been some incentive for the ad tech industry to reach a consensus with privacy advocates and other stakeholders—some reason why a failure to reach a negotiated agreement would be a worse outcome for the industry," said Arvind Narayanan, a professor at Princeton University who was one of the technologists at the table."
– Reported by Gizmodo
There you have it, "Do Not Track" failed to keep you private because Ad Tech industries failed to reach consensus or had any proper incentive to keep you private, nor any government regulation to force companies to honor "DNT"… your data is money for them. So they need you to keep planting it so they can harvest it.
Even if there is something out there saying you're not being tracked, and if it's a free service, guess what, you're being tracked. By some level you always are. Deleting cookies, disabling GPS, whatever it is, it hasn't stopped the tracking. But you can keep making layers to protect yourself and keep pushing companies to stop data mining.
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