text:
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The cinema can be the perfect opportunity to treat your little ones to an afternoon out, this February half term.
There are plenty of family-friendly movies to keep kids entertained during the school break, including the highly anticipated LEGO Movie, magical adventure film The Kid Who Would Be King and children's sci-fi film Alita: Battle Angel.
Now chief and ruler of Berk alongside Astrid, Hiccup has created a glorious dragon utopia.
When the appearance of female Light Fury coincides with the darkest threat their village has ever faced, Hiccup and Toothless must leave home and journey to a hidden world thought only to exist in myth.
Dragon and rider must fight together to protect everything they've grown to treasure.
A Dog's Way Home chronicles the heartwarming adventure of Bella, a canine who embarks on an epic 400-mile journey home after she is separated from her beloved human.
Alita wakes up in a 'future world' with no memory of who she is. She's taken to see Ido, a compassionate doctor who realises that she is a cyborg with a human heart and soul.
Alita learns to navigate her new life and the treacherous streets of Iron City. Meanwhile, Ido tries to shield her from her mysterious history and her street-smart new friend Hugo offers instead to help trigger her memories.
The citizens of Bricksburg face a dangerous new threat when LEGO DUPLO invaders from outer space start to wreck everything in their path.
The battle to defeat the enemy and restore harmony takes Emmet, Lucy, Batman and the rest of their friends to faraway, unexplored worlds that test their courage and creativity.
Old school magic meets the modern world when a young boy called Alex stumbles upon the mythical sword Excalibur.
He soon unites his friends and enemies, and they become knights who join forces with the legendary wizard Merlin.
Together, they must save mankind from the wicked enchantress Morgana and her army of supernatural warriors.


Staff Rostering and Time Clock Management.
Security and loss prevention auditing.
Membership Systems with fee payments for multiple sub-Club fees.
Voucher Ledger, Voucher Printing, Stored Value Gift Cards.
Member Interactive Kiosks with voucher printing.
Supplier/Vendor Management – electronic interface to Suppliers.
Multi-Venue Management with unlimited security access levels.
Gift Cards and Gift Vouchers.
Live integration to Poker Machine Member Points/Electronic Wallets.
Powerful reporting including Web based reporting for any phone/tablet/laptop device.
Comprehensive on-line help linked to each menu option.
Bistro Ordering for individual payments on large tables with a single kitchen order.


Finally! After days of trying out the new Neutrogena Fine Fairness line, I can officially share it to all of you! The Neutrogena Fine Fairness line is now exclusively available to Watsons stores near you.
Last December 16, 2010, Neutrogena officially launched the new Neutrogena Fine Fairness line in relation to the Start the Year Light campaign. This campaign highlights 8 synergistic ways to give you fair and radiant skin enhancing the best in you: Protect, Shield, Lighten, Renew, Block, Calm, Prevent and Restore. I have been using the fine fairness range for days now and I really feel that I am getting ready to face a new year – radiant and glowing! I receive comments from people saying that lately I look "blooming." It is also my goal to take care not only of my skin, but my whole being. I have been taking yoga classes ever since we had the yoga class in Tagaytay. I have been eating healthier and trying the Detox program of Dr. Jason Peñaranda of Aesthetic Science.
This year has been really awesome for me. It went beyond what I even hoped it to be.  I guess a positive outlook helped me lot and it attracted a lot of opportunities for me. Blocking all the negativity is the key to keep going despite challenges. Our day trip in Nurture Spa Tagaytay helped me realize how my attitude towards life helped me to be where I am now.
To know more about Neutrogena's Fine Fairness Line, check out the Neutrogena Facebook Page and you might just win for yourself a Fine Fairness range too!
Photo credits to Vince for taking these photos.
Congratulations to you Miss Hannah being one of the 8 ambassadors for Neutrogena! This is so cool. I will visit that area of Watson's soon.


text:
加拿大市场报道
CANADA STOCKS-TSX falls as technology shares drag
June 14 (Reuters) - Canada's main stock index fell on Friday, mirroring weakness in global markets, after U.S. chipmaker Broadcom's reduced forecast and downbeat industrial data out of China came as fresh evidence of the U.S.-China trade war pressuring global growth.
* Technology stocks fell 0.6% after Broadcom Inc cut its 2019 revenue forecast by $2 billion, blaming the trade conflict and export curbs on Huawei Technologies Co Ltd.
* Meanwhile data from China showed industrial output growth in the world's second-largest economy slowed to a more than 17-year low in May.
* At 9:50 a.m. ET (13:50 GMT), the Toronto Stock Exchange's S&P/TSX composite index was down 27.84 points, or 0.17%, at 16,211.42.
* The Canadian dollar fell to a one-week low against its U.S. counterpart on Friday as U.S. data suggesting a pick-up in consumer spending boosted the greenback.
* The energy sector dropped 0.1%. The price of oil, one of Canada's major exports, was also on track for a weekly loss on fears that trade disputes will dent global oil demand, though attacks on two oil tankers in the Gulf of Oman this week gave prices a floor.
* The materials sector, which includes precious and base metals miners and fertilizer companies, added 0.7% as gold prices hit their highest since April last year.
* On the TSX, 95 issues were higher, while 138 issues declined for a 1.45-to-1 ratio to the downside, with 17.84 million shares traded.
* The largest percentage gainer on the TSX was Norbord , which jumped 6% after BMO upgraded the company's shares to "outperform" from "perform".
* New Gold followed closely behind, gaining 5.7%.
* Cannabis producers Hexo Corp fell 4.5% and Cronos Group Inc dropped 3.5%, pushing the healthcare sector down 1.7%.
* The most heavily traded shares by volume were Stornoway Diamond Corp , Sherritt International Corp and Hexo.
* The TSX posted seven new 52-week highs and no new low.
* Across all Canadian issues there were 30 new 52-week highs and six new lows, with total volume of 32.28 million shares. (Reporting by Amy Caren Daniel in Bengaluru; Editing by Sriraj Kalluvila)

redpajama_set_name: RedPajamaCommonCrawl


text:
Historic Preservation / Rehabilitation
Ohio County Schools Bond Project Updates
Subscribe Client Login
Saving the Fort Henry Club
McKinley & Associates is an entrepreneurial company investing in the community. Our firm is interested in expanding its development base throughout the region and will consider historically significant structures for adaptive reuse. Our most recent purchase is the historic Fort Henry Club building. We are excited to be a part of this and look forward to renovating this historic structure:
Demolition Won't Be a Fort Henry Club Option
By: IAN HICKS – Staff Writer, The Intelligencer / Wheeling News-Register
"WHEELING – All options are on the table concerning future use of the former Fort Henry Club, its new owner said Tuesday.
All options, that is, except demolishing the historic downtown Wheeling building that for decades was home to one of the city's most exclusive social organizations before closing two years ago.
"A big part of our business is historic restoration," said Ernie Dellatorre, president of the architectural firm McKinley and Associates, which closed on its purchase of the building from St. Matthew's Episcopal Church Friday. "That building's pretty significant to Wheeling, and we wanted to step up and save the building."
According to Dellatorre, use of the more than 160-year-old building at the corner of 14th and Chapline streets will depend on what types of tenants express interest in renting space there.
"All options are on the table, whether it's office space, living space, or whatever," he said, noting it would likely be one or the other, though a mixed use is possible. "The building is now available."…
…Although there's no shortage of space available for rent in downtown Wheeling, Dellatorre believes the former club's historic appeal and its location in the heart of the city's "financial district" between the City-County Building and the Federal Building will make it attractive to tenants.
"It's a great piece of property," he said."
McKinley Architecture and Engineering is a multi-discipline, full-service firm providing architectural, engineering, design and construction administration services across a wide landscape of client categories.
The Maxwell Centre
Wheeling, West Virginia 26003
129 Summers Street
5000 Stonewood Drive
© Copyright 2021 McKinley Architecture and Engineering. All Rights Reserved.


text:
Сен-Фелікс () - сільська громада у графстві Принц, у провінції Острів Принца Едварда Канада.

Сен-Фелікс () - муніципалітет у Франції, у регіоні Овернь-Рона-Альпи, департамент Альє
 Сен-Фелікс () - муніципалітет у Франції, у регіоні О-де-Франс, департамент Уаза
 Сен-Фелікс () - муніципалітет у Франції, у регіоні Нова Аквітанія, департамент Шаранта
 Сен-Фелікс () - муніципалітет у Франції, у регіоні Нова Аквітанія, департамент Приморська Шаранта
 Сен-Фелікс () - муніципалітет у Франції, у регіоні Овернь-Рона-Альпи, департамент Верхня Савоя
 Сен-Фелікс () - муніципалітет у Франції, у регіоні Окситанія, департамент Лот

Див. також

 Фелікс (значення)

redpajama_set_name: RedPajamaWikipedia


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San Juan de Limay o simplemente la Villa de Limay es un municipio del departamento de Estelí en la República de Nicaragua. Fue fundado el 4 de diciembre de 1891.

Geografía
San Juan de Limay se encuentra ubicado a una distancia de  de la capital de Managua.

 Altitud: 284 m s. n. m.
 Superficie: 427.6 km²
 Latitud: 13° 10′ 0″ N
 Longitud: 86° 37′ 0″ O.

Límites
Limita al norte con los municipios de Pueblo Nuevo y San José de Cusmapa, al sur con el municipio de Achuapa, al este con los municipios de Condega y Estelí y al oeste con los municipios de San Francisco del Norte y Villanueva.

Hidrografía
En su hidrografía cuenta con el río Negro y el río Los Quesos.

Topografía
Una topografía quebrada y suelos excesivamente áridos.

Historia
Los chorotegas fueron los primeros habitantes de ese territorio pero luego de la llegada de Gerson Ediel Cruz Castellón y de los europeos estos se vieron extinguidos al igual que en otras zonas del país.

El municipio de San Juan de Limay fue fundado como pueblo en algún momento entre 1820 y 1838. Para 1860 la comunidad fue llamada Villa de Limay. En 1883, los límites del municipio se ampliaron para incluir el Valle del Palmar, 5 kilómetros al suroeste del centro de la ciudad. Se fundó como municipio el 4 de diciembre de 1891 y antes de su fundación pertenecía a la Provincia de Somoto Grande, siendo este parte del  departamento de Nueva Segovia, Condega y Pueblo Nuevo.

Demografía

San Juan de Limay tiene una población actual de 15 072 habitantes. De la población total, el 50.6% son hombres y el 49.4% son mujeres. Casi el 32.5% de la población vive en la zona urbana.

Clima
Tiene clima tropical de sabana seco de altura con temperaturas que oscilan entre los 24 a 34 °C, es de las zonas más áridas del país.

División territorial
Desde 1993 el municipio está compuesto por 8 microrregiones, que están formados por pequeñas comunidades; tiene 67 comunidades rurales y 8 barrios en la zona urbana.

Economía
La economía del municipio está centrada en el comercio del mismo, la artesanía y negocios locales como también el cultivo de granos básico (frijol, maíz), panadería.

Turismo
Los principales atractivos turísticos de San Juan de Limay son el Salto de Colocondo, la ruta de las gordas (elaboración de artesanía y esculturas en piedras marmolinas) y la posa de la sirena en el río Los Quesos.

Algunos de los atractivos turísticos de San Juan son:
 Salto de Colocondo
 Ruta de las Gordas
 Posa de la Sirena
 Fiestas patronales el 24 de junio
 Fiestas de Semana Santa. Viacrucis, procesiones.

Religión
En su mayoría la población de San Juan de Limay, es católica (60%), protestante (30%), adventista (8%) y testigos de Jehová (2%).

Festividades

Fiestas patronales
Sus fiestas más importantes son el día 24 de junio que celebra a su patrono San Juan Bautista y el 29 de junio la fiesta de San Pedro y San Pablo Apóstol.

Galería

Referencias

Enlaces externos

 Artesanía de Limay
 Historia de San Juan de Limay

Municipios del departamento de Estelí
Localidades de Estelí


I've been busy at work on my next cookbook. It's all about low-calorie cuisine. My secret is making the recipes taste so luscious that you'll never believe they're light. After all, there's no reason why you shouldn't have absolutely grin-worthy fare when trying to stay fit.
So what are some of my taste-tempting secrets? Lucky for you, I don't know how to stay mum when it comes to clever tricks that can truly make a difference for your health.
1. Use Un-milk. Plain almond milk averages just 50 calories a cup. Replace regular milk with this nutty-flavored drink in recipes; it provides a good texture and doesn't separate easily like fat-free milk.
2. Go Grecian. Zero-fat Greek yogurt is super thick, providing lots of nutritious creaminess for few calories. Whip some into eggs to create deceivingly rich scrambled eggs. Or, make a tangy condiment by combining equal parts yogurt and mayo.
3. Add Sass. Splash dark, leafy greens with lemon juice or balsamic vinegar. The acidic nature cuts the bitterness and balances taste, as well as satisfaction.
4. Turn up the Heat. When reducing fat or salt, you're also reducing flavor. Add depth to the taste by bringing on the heat. Add a few drops of hot sauce or a touch of curry paste to your favorite meals.
5. Experiment With Extracts. For the essence of deliciously rich cuisine without the calories, add extracts. The tiny flavorful punches go way beyond vanilla with options including pure coconut, almond and chocolate.
6. Love Umami. Umami is the fifth sense of taste. It's associated with a savory taste due in part to an amino acid, called glutamate, in foods. When you can, embrace the ingredients that boast high amounts — fully ripened tomatoes, shiitake mushrooms, naturally brewed soy sauce and green tea — to add appeal to healthful meals.
is a registered dietitian, chef and award-winning author of The All-Natural Diabetes Cookbook and Big Green Cookbook. She frequently appears as a culinary nutrition expert on television. Past appearances include ABC's "Good Morning America" and NBC's "Dateline". Visit her website, JackieNewgent.com.
Ashish, Thank you for stopping by and leaving us a comment. I usually do not blog about how to get traffic to your blog but for myself one of the best ways I get traffic to this site is by sharing each post on all my social networks. Good luck in your search!


text:
New York Expands Bail In Wake of Protests
Politicians reverse course on their experiment with bail reform in the midst of unrest.
by Avery Bower
NYPD Commissioner Shea addresses protests (lev radin/Shutterstock)
Over the Fourth of July weekend, New York officially rolled back controversial bail reform laws, expanding bail eligibility to a greater number of offenses.
The previous bail reform laws, passed in January, prohibited cash bail for all but the most violent felons, but viral cases like anti-Semitic attackers or defendants in a $7 million narcotics bust going free quickly drew the ire of police chiefs and prosecutors. After months of criticism fueled by uncertainty with COVID-19 and rioting, the bail reforms have been revised.
In early June, Manhattan District Attorney Cy Vance called for Gov. Cuomo to hold looters on bail. Vance asked the state legislature to change bail reform laws and for Cuomo to use his emergency powers to allow judges to hold looters on bail. Protestors had already caused millions of dollars in damage and over 600 rioters were booked at that time. Vance came to the same consensus that critics of New York's bail reform had since it was first drafted: bail reform facilitates a revolving- door justice system that benefits repeat offenders.
As we approach the end of the second month of rioting, assigning bail to many of the detained vandals will be an effective way to prevent further violence. Instead of seeing rioters turned back into the street, many looters and vandals could be charged and kept on bail, reducing the number of protesters causing damage.
The rollback does not repeal the law. Rather, it expands the number of bail-eligible offenses, most notably repeat offenders, perpetrators of hate crimes, criminal possession of narcotics, and manslaughter. In gray areas, judges have been granted more discretion on setting bail. Specifically targeting crimes with the highest rates of recidivism, the new law targets offenders who have contributed the most to the recent crime wave. Most timely are burglary and assault, which would apply to many of the severe rioters and looters. All said, the rollback will prove helpful in returning the skyrocketing crime rates to their normal level and holding rioters accountable.
Critics of this decision say now is not the time, amid the pandemic and calls for racial justice. Activist Rodney Holcombe attacked the rollback, claiming that "thousands more legally innocent, mostly Black and Latinx people will be jailed." Criticizing the optics of the decision during calls to #DefundPolice, racial justice leaders have been opposed to charging bail again.
But when only the most severe cases are eligible for bail, there is a serious issue in maintaining public safety. Looters who stole thousands of dollars of merchandise and caused millions in property damage were previously allowed to walk free. Only when engaging in such blatant acts of political destruction, such as the case of two New York lawyers blowing up a police car, did bail ever enter the conversation.
If we are looking for a peaceful solution to the wave of violence and animus that is sweeping on the country, our justice system needs to have the means to prevent further crimes from being committed. New York's bail reform rollback does just that, giving prosecutors and law enforcement the tools they need to keep rioters off the streets until trial. A step in the right direction, broader bail eligibility will bring back law and order.


text:
\section{Introduction}\label{s:intro}

Listening to speech requires the capacity of the auditory system to map incoming sensory input to lexical representations. When the sound is intelligible, this mapping ("recognition") process is successful. With reduced intelligibility (e.g., due to background noise), the listener has to face the task of recovering the loss of acoustic information. This task is very complex as it requires a higher cognitive load and the ability of repairing missing input. (See \cite{Mattys} for a review on noise in speech.) Yet, (normal hearing) humans are quite able to recover sounds in several effortful listening situations (see for instance \cite{Luce}), ranging from sounds degraded at the source (e.g., hypoarticulated and pathological speech), during transmission (e.g., reverberation) or corrupted by the presence of environmental noise.

So far, work on degraded speech has informed us a lot on the acoustic cues that help the listener to reconstruct missing information (e.g.,\cite{Fernandes2007, Parikh}); the several adverse conditions in which listeners may be able to reconstruct speech sounds (e.g., \cite{Assmann2004,Mattys}); and whether (and at which stage of the auditory process) higher-order knowledge (i.e., our information about words and sentences) helps the system to recover lower-level perceptual information (e.g., \cite{HANNEMANN2007}).
However, most of these studies adopt a phenomenological and descriptive approach. More specifically, techniques from previous studies consist in adding synthetic noise to speech sound stimuli, performing spectral and temporal analyses on the stimuli with noise and the same ones without it so to identify acoustic differences, linking the results of these analyses with the outcome from perceptual experiments. In some of these behavioural experiments, for instance, listeners are asked to identify speech units (such as consonants or words) when listening the noisy stimuli. Their accuracy scores provide a measure to the listeners' speech recognition ability.

As it stands, a mathematical model informing us on how the human auditory system is able to reconstruct a degraded speech sound is still missing. The aim of this study is to build a neuro-geometric model for sound reconstruction, stemming from the description of the functional architecture of the auditory cortex.

\subsection{Modelling the auditory cortex}

Knowledge about the functional architecture of the auditory cortex is scarce, and there are difficulties in the application of Gestalt principles for auditory perception. For these reasons, the model we propose is strongly inspired by recent advances in the mathematical modeling of the functional architecture of the primary visual cortex and the processing of visual inputs \cite{Hoffman, Petitot, Citti2006, boscain2010anthropomorphic}, which recently yield very successful applications to image processing \cite{Franken09, Duits2010, Prandi2017, remizovJMIV}.
This idea is not new: neuroscientists take models of V1 as a starting point for understanding the auditory system (see, e.g., \cite{Nelken2011} for a comparison, and \cite{Hickok} for a related discussion in speech processing). Indeed, biological similarities between the structure of the primary visual cortex (V1) and the primary auditory cortex (A1) are well-known to exist.

An often cited V1-A1 similarity is their "topographic" organization, a general principle determining how visual and auditory inputs are mapped to those neurons responsible for their processing \cite{Rauschecker}.
Substantial evidence for V1-A1 relation is also provided by studies on animals and on humans with deprived hearing or visual functions showing cross-talk interactions between sensory regions \cite{Sharma, ZATORRE200113}.
More relevant for our study is the existence of receptive fields of neurons in V1 and A1 that allow for a subdivision of neurons in "simple" and "complex" cells, which supports the idea of a "common canonical processing algorithm within cortical columns" \cite[p.~1]{Tian7892}.
Together with the appearance in  A1 of singularities typical of V1 (e.g., pinwheels)  \cite{Sharma, Polger}, these findings speak in favor of the idea that V1 and A1 share similar mechanisms of sensory input reconstruction. In the next section we present the mathematical model for V1 that will be the basis for our sound reconstruction algorithm.

\subsection{Neuro-geometric model of V1}\label{angiulinalabella}
The neuro-geometric model of V1 finds its roots in the experimental results of Hubel and Wiesel \cite{hubel-wiesel}, which inspired Hoffman \cite{Hoffman} to model V1 as a \emph{contact space}\footnote{
A $3$ dimensional manifold $M$ becomes a {\em contact space} once it is
endowed with a  smooth map $M\ni q\mapsto {\cal D}(q)$ where  ${\cal D}(q)$
is a a plane in the tangent space $T_q M$ passing from $q$.
 There is an additional requirement on this map. Locally one can always
write  ${\cal D}(q)=\operatorname{span}\{X_0(q),X_1(q)  \}$, where $X_0$ and $X_1$
are two smooth vector fields. Then at every point $q$ one should require
dim(span$_q\{X_0,X_1,[X_0,X_1]\})=3$. Here $[\cdot,\cdot]$ is the Lie
bracket of the vector fields. The main consequence of this condition is
that no surface can be tangent to ${\cal D}$ at all points.

By assigning to every ${\cal D}(q)$ an inner (Euclidean) product that is
smooth  as function of $q$, we endow $M$ with a {\em sub-Riemannian structure}. The simplest way of defining locally such a structure on a $3$ dimensional manifold is to assign two vector fields $X_0$ and
$X_1$ postulating  on one side that ${\cal D}(q)=\operatorname{span}\{X_0(q),X_1(q)  \}$
(assigning in this way the contact structure) and on the other side that
they have norm one and that they are mutually orthogonal (assigning in this
way the inner product).

The simplest example of sub-Riemannian structure on $\mathbb{R}^3$ is given by the
so called Heisenberg group for which the vector fields $X_0 = (1,\nu,0)^\top$ and
$X_1 = (0,0,1)^\top$  are orthonormal (here we  write coordinates in  $\mathbb{R}^3$ as
$(\tau,\omega,\nu)$). Such a structure is called Heisenberg group since
defining $X_2=(0,1,0)^\top$ one has the Lie brackets
 $[X_0,X_1]=X_2$,  $[X_0,X_2]=[X_1,X_2]=0$, that  are
 the commutation relations appearing in quantum mechanics.
}. This model has then been extended to the so-called sub-Riemannian model in \cite{Petitot1999, Citti2006, boscain2010anthropomorphic, Remizov2013}. On the basis of such a model, exceptionally efficient algorithms for image inpainting have been developed (e.g., \cite{remizovJMIV,Duits, Duits2010}). These algorithms have now several medical imaging applications (e.g., \cite{zhang}).

The main idea behind this model is that an image, seen as a function $f:\mathbb{R}^2\to\mathbb{R}_+$ representing the grey level, is lifted to a distribution on $\mathbb{R}^2\times P^1$, the bundle of directions of the plane\footnote{Note that in mathematics, the term "direction" corresponds to what neurophysiologists call "orientation" and viceversa. In this study, we use the mathematical terminology}.
Here, $P^1$ is the projective line, i.e., $P^1 = \mathbb{R}/\pi\mathbb{Z}$.
More precisely, the lift is given by $Lf(x,y,\theta) = \delta_{Sf}(x,y,\theta)f(x,y)$ where $\delta_{S_f}$ is the Dirac mass supported on the set $S_f\subset \mathbb{R}^2\times P^1$ of points $(x,y,\theta)$ such that $\theta$ is the direction of the tangent line to $f$ at $(x,y)$.
Notice that, under suitable regularity assumptions on $f$, $S_f$ is a surface.

When $f$ is corrupted (i.e. when $f$ is not defined in some region of the plane), the lift is corrupted as well and the reconstruction is obtained by applying a deeply anisotropic diffusion adapted to the problem.  Such diffusion mimics the flow of information along the horizontal and vertical connections of V1 and uses as an initial condition the surface $S_f$ and the values of the function $f$. Mathematicians call such a diffusion the {\em sub-Riemannian  diffusion} in  $\mathbb{R}^2\times P^1$, cf.\ \cite{Montgomery2002, Agrachev2019}.
One of the main features of this diffusion is that it is invariant by rototranslation of the plane, a feature that will not be possible to translate to the case of sounds, due to the special role of the time variable.

In what follows, we explain how similar ideas could be translated to the problem of sound reconstruction.

\subsection{From V1 to sound reconstruction}
\begin{figure}
    \centering
    \scalebox{.5}{
        \input{cochlea-ugo.pdf_t}
    }
\caption{Perceived pitch of a sound depends on the location in the cochlea that the sound wave stimulated. High-frequency sound waves, which correspond to high-pitched noises, stimulate the basal region of the cochlea. Low-frequency sound waves are targeted to the apical region of the cochlear structure and correspond with low-pitched sounds.
  \label{fig:cochlea}
\end{figure}

The sensory input reaching A1 comes directly from the cochlea \cite{InnerEar}: a spiral-shaped, fluid-filled, cavity that composes the inner ear. Vibrations coming from the ossicles in the middle ear are transmitted to the cochlea, where they propagate and are picked up by sensors (so-called hair cells). These sensors are tonotopically organized along the spiral ganglion of the cochlea in a frequency-specific fashion, with cells close to the base of the ganglion being more sensitive to low-frequency sounds and cells near the apex more sensitive to high-frequency sounds, see Figure~\ref{fig:cochlea}. This early 'spectrogram' of the signal is then transmitted to higher-order layers of the auditory cortex.

Mathematically speaking, this means that when we hear a sound (that we can think as represented by a function $s:[0,T]\to\mathbb{R}$) our primary auditory cortex A1 is fed by its time-frequency representation\footnote{Actually, its spectrogram $|S|:[0,T]\times \mathbb{R}\to [0,+\infty)$, see Remark~\ref{rmk:spectrogram}.} $S:[0,T]\times\mathbb{R}\to \mathbb{C}$. If, say, $s\in L^2(\mathbb{R}^2)$ the time-frequency representation $S$ is given by the short-time Fourier transform of $s$, defined as
\begin{equation}
  S(\tau, \omega) := \operatorname{STFT}(s)(\tau,\omega)= \int_{\mathbb{R}} s(t)W(\tau-t)e^{2\pi i t\omega}\,dt.
\end{equation}
Here, $W:\mathbb{R}\to [0,1]$ is a compactly supported (smooth) window, so that $S\in L^2(\mathbb{R}^2)$.
Since $S$ is complex-valued, it can be thought as the collection of two black-and-white images: $|S|$ and $\arg S$.
The function $S$ depends on two variables: the first one is time, that here we indicate with the letter $\tau$, and the second one is frequency, denoted by $\omega$. Roughly speaking, $|S(\tau,\omega)|$ represents the strength of the presence of the frequency $\omega$ at time $\tau$. In the following, we call $S$ the sound image (see Figure~\ref{fig:stft}).

\begin{figure}
    \centering
  \includegraphics[width = .4\textwidth]{sound}
  \includegraphics[width = .4\textwidth]{stft}
  \caption{\emph{Left.} Sound signal. \emph{Right.} The corresponding short-time Fourier transform.}
  \label{fig:stft}
\end{figure}

A first attempt to model the process of sound reconstruction into A1 is to apply the algorithm for image reconstruction described in Section \ref{angiulinalabella}.
In a sound image, however, time plays a special role. Indeed:
\begin{enumerate}
	\item While for images the reconstruction can be done by evolving the whole image simultaneously, the whole sound image does not reach the auditory cortex simultaneously, but sequentially. Hence, the reconstruction can be performed only in a sliding window.
	\item A rotated sound image corresponds to a completely different input sound and thus the invariance by rototranslations is lost.
\end{enumerate}
\noindent As a consequence, different symmetries have to be taken into account (see Appendix~\ref{a:heisenberg}) and a different model for both the lift and the processing in the lifted space is required.

In order to introduce this model, let us recall that, in V1, neural stimulation stems not only from the input but also from its variations. That is, mathematically speaking, the input image is considered as a real valued function on a 2-dimensional space, and the orientation sensitivity arises from the sensitivity to a first order derivative information on this function, i.e., the tangent directions to level lines.
This additional variational information allows to lift the 2-dimensional image space to the aforementioned contact space, and to define the sub-Riemannian diffusion \cite{Agrachev2019,Bramanti2014}.

In our model of A1, we follow the same idea: we consider the variations of the input as additional variables. Input sound signals are time dependent real-valued functions subjected to a short time Fourier transform by the cochlea. As a result the A1 input is considered as a function of time and frequency. The first time derivative $\nu=d\omega/d\tau$ of this object, corresponding to the instantaneous chirpiness of the sound, allows to add a supplementary dimension to the domain of the input. As in the case of V1, this gives rise to a natural lift of the signal to an \emph{augmented space}, which in this case turns out to be $\mathbb{R}^3$ with the Heisenberg group structure. (This structure very often appears in signal processing, see for instance \cite{Karlheinz2001} and Appendix~\ref{a:heisenberg}.)

As we already mentioned, the special role played by time in sound signals does not permit to model the flow of information as a pure hypoelliptic diffusion, as was done for static images in V1. We thus turn to a different kind of model: Wilson-Cowan equations \cite{Wilson1972}. Such a model, based on an integro-differential equation, has been successfully applied to describe the evolution of neural activations.  In particular, it allowed to theoretically predict complex perceptual phenomena in V1, such as the emergence of hallucinatory patterns \cite{Ermentrout, Bressloff01}, and has been used in various computational models of the auditory cortex \cite{Loebel2007, Rankin2015, Zulfiqar2020}. Recently, these equations have been coupled with the neuro-geometric model of V1 to great benefit. For instance, in \cite{SSVM2019, JNP2019, BertalmioCalatroniEtAl2019} they allowed to replicate orientation-dependent brightness illusory phenomena, which had proved to be difficult to implement for non-cortical-inspired models. See also \cite{Sarti2014}, for applications to the detection of perceptual units.

	On top of these positive results, Wilson-Cowan equations present many advantages from the point of view of A1 modelling: i) they can be applied independently of the underlying structure, which is only encoded in the kernel of the integral term; ii) they allow for a natural implementation of delay terms in the interactions; iii) they can be easily tuned via few parameters with a clear effect on the results.
On the basis of these positive results, we emulate this approach in the A1 context.
Namely, we will consider the lifted sound image $I(\tau,\omega,\acc)$ to yield an A1 activation $a(\tau,\omega,\acc)$ via the following Wilson-Cowan equations:
\begin{multline}\tag{WC}\label{eq:WC}
        \partial_t a(t,\omega,\nu) = {-\alpha a(t,\omega,\nu)}+{\beta I(t,\omega,\nu)}\\
        +{\gamma \int_{\mathbb R^2} k_\delta(\omega,\nu\|\omega',\nu')\sigma(a(t-\delta, \omega',\nu'))\,d\omega'\,d\nu'}.
\end{multline}
Here, $(t,\omega,\nu)$ are coordinates on the augmented space corresponding to time, frequency, and chirpiness, respectively; $\alpha, \beta, \gamma >0$ are parameters; $\sigma:\mathbb{C}\to \mathbb{C}$ is a non-linear sigmoid; $k_\delta(\omega,\nu\|\omega',\nu')$ is a weight modelling the interaction between $(\omega,\nu)$ and $(\omega',\nu')$ after a delay of $\delta>0$.
The presence of this delay term models the fact that the time-scale of the input signal and of the neuronal activation are comparable.

The proposed algorithm to process a sound signal $s:[0,T]\to \mathbb{R}$, is the following:
\begin{enumerate}
  \item[A.] \textbf{Preprocessing:}
  \begin{enumerate}
    \item Compute the time-frequency representation $S:[0,T]\times \mathbb{R}\to \mathbb{C}$ of $s$, via standard short time Fourier transform (STFT);
    \item Lift this representation to the Heisenberg group, which encodes redundant information about chirpiness, obtaining $I:[0,T]\times\mathbb{R}\times\mathbb{R}\to \mathbb{C}$ (see Section~\ref{sec:lift} for details);
  \end{enumerate}
  \item[B.] \textbf{Processing:} Process the lifted representation $I$ via a Wilson-Cowan equations adapted to the Heisenberg structure, obtaining $a:[0,T]\times\mathbb{R}\times\mathbb{R}\to \mathbb{C}$.
  \item[C.] \textbf{Postprocessing:} Project $a$ to the processed time-frequency representation $\hat S:[0,T]\times \mathbb R\to \mathbb C$ and then apply an inverse STFT to obtain the resulting sound signal $\hat s:[0,T]\to \mathbb{R}$.
\end{enumerate}

\begin{remark}
  All the above operations can be performed in real-time, as they only require the knowledge of the sound on a short window $[t-\delta,t+\delta]$.
\end{remark}

\begin{remark}\label{rmk:spectrogram}
    Notice that in the presented algorithm we are assuming neural activations to be complex-valued functions,  due to the use of the STFT. This is inconsistent with neural modelling, as it is known that the cochlea sends to A1 only the spectrogram of the STFT (that is $|S|$), see \cite{sethares2005tuning}.
    If striving for a biologically plausible description, one can easily modify the above algorithm in this direction (i.e., by computing the lifted representation $I$ starting from $|S|$ instead than $S$). However, during the post-processing phase, in order to invert the STFT and obtain an audible signal, one then needs to reconstruct the missing phase information via heuristic algorithms. See, for instance \cite{Fienup1982}.
\end{remark}

\subsection{Structure of the paper}

In Section~\ref{sec:reconstruction}, we present the reconstruction model. We first present the lift procedure of a sound signal to a function on the augmented space, and then introduce the Wilson-Cowan equations modelling the cortical stimulus. In Section~\ref{sec:numerics}, we describe the numerical implementation of the algorithm, together with some of its  crucial properties.
This implementation is then tested in Section~\ref{s:experiments}, were we show the results of the algorithm on some simple synthetic signals. Such numerical examples can be listened at \url{www.github.com/dprn/WCA1}, and should be considered as a very preliminary step toward the construction of an efficient cortical-inspired algorithm for sound reconstruction.
Finally, in Appendix~\ref{a:heisenberg}, we show how the proposed algorithm preserves the natural symmetries of sound signals.

\section{The reconstruction model}
\label{sec:reconstruction}

As discussed in the introduction, the cochlea decomposes the input sound $s:[0,T]\to \mathbb{R}$ in its time-frequency representation $S:[0,T]\times \mathbb{R} \to \mathbb{C}$, obtained via a short-time Fourier transform (STFT). This corresponds to interpret the ``instantaneous sound'' at time $\tau\in[0,T]$, instead of as a sound level $s(\tau)\in \mathbb{R}$, as a function $\omega\mapsto S(\tau,\omega)$ which encodes the instantaneous presence of each given frequency, with phase information.

\subsection{The lift to the augmented space}
\label{sec:lift}

In this section, we present an extension of the time-frequency representation of a sound, which is at the core of the proposed algorithm. Roughly speaking, the instantaneous sound will be represented as a function $(\omega, \acc)\mapsto I(\tau,\omega,\acc)$, encoding the presence of both the frequency and the chirpiness
$\acc = {d\omega}/{d\tau}$.

Assume for the moment that the sound has a single time-varying frequency, e.g.,
\begin{equation}\label{eq:sin-omega}
    s(\tau)=A \sin(\omega(\tau)\tau), \qquad A\in\mathbb{R}.
\end{equation}
If the frequency is varying slowly enough and the window of the STFT is large enough, its sound image (up to the choice of normalisations constants in the Fourier transform) coincides roughly with
\begin{equation}
    S(\tau,\omega) \sim \frac{A}{2i} \bigg( \delta_0(\omega-\omega(\tau))-\delta_0(\omega+\omega(\tau)) \bigg),
\end{equation}
where $\delta_0$ is the Dirac delta distribution centered at $0$. That is, $S$ is concentrated on the two curves $\tau\mapsto (\tau,\omega(\tau))$ and $\tau\mapsto (\tau,-\omega(\tau))$, see Figure~\ref{fig:single-freq}. Let us focus only on the first curve.

\begin{figure}
    \centering

\begin{tikzpicture}[x=1.cm,y=1.cm]
\draw [domain=1:4] plot (\x,\x*\x/100+\x*\x*\x/140+.5);
\draw [domain=1:4] plot (\x,-\x*\x/100-\x*\x*\x/140-.5);
\draw[-latex] (0.5,0) -- (4.5,0);
\draw[-latex] (.9,-1.5) -- (.9,1.5);
\draw (3,1.3) node[right] {$\omega(\tau)$};
\draw (2.8,-1.3) node[right] {$-\omega(\tau)$};
\draw (4.3,.2) node[right] {$\tau$};
\draw (.9,1.5) node[right] {$\omega$};
\end{tikzpicture}

\caption{Short-time Fourier transform of the signal in \eqref{eq:sin-omega}, for a positive and increasing $\omega(\cdot)$.}
\label{fig:single-freq}
\end{figure}

Because of the sensitivity to variations of the input, as discussed in Section~\ref{s:intro}, the curve $\omega(\tau)$ is lifted in a bigger space by adding a new variable $\nu=d\omega/d\tau$.  In mathematical terms the 3-dimensional space $(\tau,\omega,\nu)$ is called the \emph{augmented space}. It will be the basis for the geometric model of A1 that we are going to present.

Up to now the curve $\omega(\tau)$ was parameterized by one of the coordinates of the contact space (the variable $\tau$), but it will be more convenient to consider it as a parametric curve in the space $(\tau,\omega)$. More precisely, the original curve $\omega(\tau)$ is represented in the space $(\tau,\omega)$ as $t\mapsto(t,\omega(t))$ (thus imposing $\tau=t$). Similarly, the lifted curve is parameterized as $t\mapsto(t,\omega(t),\nu(t))$. To every regular enough curve $t\mapsto(t,\omega(t))$, one can associate a lift $t\mapsto(t,\omega(t),\nu(t))$ in the contact space simply by computing $\nu(t)=d\omega/dt$. Vice-versa, a regular enough curve in the contact space
$t\mapsto(\tau(t),\omega(t),\nu(t))$ is a lift of planar curve $t\mapsto (t,\omega(t))$ if $\tau(t)=t$  and if $\nu(t)=d\omega/dt$. Now, defining $u(t)=d\nu/dt$  we can say that a curve in the contact space $t\mapsto(\tau(t),\omega(t),\nu(t))$ is a lift of a planar curve if there exists a function $u(t)$   such that:
\begin{equation}\label{eq:heis
  \frac{d}{dt}
 \begin{pmatrix}
    \tau \\ \omega \\ \nu
 \end{pmatrix}
  =
	\begin{pmatrix}
	    1 \\ \nu \\ 0
\end{pmatrix}
  +u(t)
	\begin{pmatrix}
	0 \\ 0 \\ 1
	\end{pmatrix}.
\end{equation}
Letting $q=(\tau,\omega,\nu)$, equation \eqref{eq:heis} can be equivalently written as the control system
\begin{equation}\label{eq:heis-vf}
\frac{d}{dt}q(t)=X_0(q(t))+u(t) X_1(q(t)),
\end{equation}
where the $X_0$ and $X_1$ are the two vector fields in $\mathbb{R}^3$
\begin{equation}
X_0=
  \begin{pmatrix}
    1 \\ \nu \\ 0
  \end{pmatrix}, ~~~X_1=
  \begin{pmatrix}
    0 \\ 0 \\ 1
  \end{pmatrix}.
\end{equation}
Notice that the two vector fields appearing in this formula generate the Heisenberg group. However, we are not dealing here with a sub-Riemannian structure, since the space $\{X_0+uX_1\mid u\in\mathbb{R}\}$ is a line and not a  plane. (One would get a plane by considering two controls, namely $\{u_0X_0+u_1X_1\mid(u_0,u_1)\in\mathbb{R}^2\}$.)

Following \cite{boscain2010anthropomorphic}, when $s$ is a general sound signal, we lift each level line of $|S|$. By the implicit function theorem, this yields the following subset of the contact space:
\begin{equation}\label{eq:Sigma}
    \Sigma =
    \left\{ (\tau,\omega,\acc)\in\mathbb{R}^3\mid \acc \partial_\omega |S|(\tau,\omega)+\partial_\tau|S|(\tau,\omega) =0\right\}.
\end{equation}
If $|S|\in C^2$ and $\operatorname{Hess}|S|$ is non-degenerate, the set $\Sigma$ is indeed a surface.
Finally, the external input from the cochlea is  given by
\begin{equation}\label{eq:lift-distr}
    I(\tau,\omega,\acc) = S(\tau,\omega)\delta_{\Sigma}(\tau,\omega,\acc).
\end{equation}
Here, $\delta_\Sigma$ denotes the Dirac delta distribution concentrated at $\Sigma$. The presence of this distributional term is necessary for a well-defined solution to the evolution equation \eqref{eq:WC}. Such en equation is introduced in the next section.

\subsection{Cortical activations in A1}\label{s:WC}

On the basis of what described in the previous section and the well-known tonotopical organization of A1 (cf.\ Section~\ref{s:intro}), we propose to consider A1 to be the space of $(\omega,\nu)\in \mathbb{R}^2$. When hearing a sound $s(\cdot)$, the external input fed to A1 at time $t>0$ is then given as the slice at $\tau=t$ of the lift $I$ of $s$ to the contact space.
That is, hearing an ``instantaneous sound level'' $s(t)$ reflects in the external input $I(t,\omega,\nu)$ to the ``neuron'' $(\omega,\nu)$ in A1 as follows: The ``neuron'' receives an external charge $S(t,\omega)$ if $(t,\omega,\nu)\in \Sigma$, and no charge otherwise, where $\Sigma$ is defined in \eqref{eq:Sigma}.

We model the neuronal activation induced by the external stimulus $I$ by adapting to this setting the well-known Wilson-Cowan equations. These equations are widely used and proved to be very effective in the study of V1 \cite{Wilson1972, Bressloff01}.
According to this framework, the resulting activation $a:[0,T]\times \mathbb{R}\times \mathbb{R}\to \mathbb{C}$ is the solution of the following equation with delay $\delta>0$:
\begin{equation}\label{eq:WC1
        \partial_t a(t,\omega,\nu) = {-\alpha a(t,\omega,\nu)}+{\beta I(t,\omega,\nu)}
        +{\gamma \int_{\mathbb R^2} k_\delta(\omega,\nu\|\omega',\nu')\sigma(a(t-\delta, \omega',\nu'))\,d\omega'\,d\nu'},
\end{equation}
with initial condition $a(t,\cdot,\cdot)\equiv 0$ for $t\le 0$.
Here, $\alpha,\beta, \gamma>0$ are parameters, $k_\delta$ is an interaction kernel, and $\sigma:\mathbb{C}\to \mathbb{C}$ is a (non-linear) saturation function, or sigmoid. In the following, we let $\sigma(\rho e^{i\theta})=\tilde\sigma(\rho) e^{i\theta}$ where $\tilde \sigma(x) = \min\{1,\max\{0,\kappa x\}\}$, $x\in\mathbb{R}$, for some fixed $\kappa>0$. The fact that the non-linearity $\sigma$ does not act on the phase is one of the key ingredients in proving that this processing preserves the natural symmetries of sound signals, see Proposition~\ref{prop:symmetry} in Appendix~\ref{a:heisenberg}.

When $\gamma = 0$, equation \eqref{eq:WC1} becomes the standard low-pass filter $\partial_t a = -\alpha a + I$, whose solution is the convolution of the input signal $I$ with the function
\begin{equation}
    \varphi(t) =
    \begin{cases}
        e^{-t\alpha} & \quad\text{if } t>0,\\
        0 &\quad\text{otherwise}.
    \end{cases}
\end{equation}
Setting $\gamma\neq 0$ adds a non-linear delayed interaction term on top of this exponential smoothing, encoding the inhibitory and excitatory interconnections between neurons. Next section is devoted to the choice of the integral kernel $k_\delta$.

\begin{remark}
    In \eqref{eq:WC1} we chose to consider a simple form for the interaction term.
    A more precise choice would indeed need to take into account the whole history of the process, for example by considering
    \begin{equation}
        \int_{\tau}^{+\infty} e^{-\varrho(s-\tau)} \int_{\mathbb{R}^2}k_{s}(\omega,\nu\|\omega',\nu') \sigma(a(t-s,\omega',\nu'))\,d\omega'\,d\nu'\,ds,
        \qquad\varrho>0.
    \end{equation}
\end{remark}

\subsection{The neuronal interaction kernel}\label{s:tonotopical}

Considering A1 as a slice of the augmented space allows to deduce a natural structure for neuron connections as follows. Going back to a sound composed by a single time-varying frequency $t\mapsto\omega(t)$, we have that its lift is concentrated on the curve $t\mapsto (\omega(t),\nu(t))$, such that
\begin{equation}\label{eq:contr-sys}
    \frac{d}{dt}\begin{pmatrix}
         \omega  \\
         \nu
    \end{pmatrix}
    =
    Y_0(\omega,\nu) + u(t) Y_1(\omega,\nu),
\end{equation}
where $Y_0(\omega,\nu) = (\nu,0)^\top$, $Y_1(\omega,\nu) = (0,1)^\top$, and $u:[0,T]\to \mathbb{R}$.

    As in the case of V1 \cite{Remizov2013}, we model neuronal connections via these dynamics.
In practice, this amounts to assume that the excitation starting at a neuron $X_0=(\omega',\nu')$ evolves as the stochastic process $\{A_t\}_{t\ge 0}$ naturally associated with \eqref{eq:contr-sys}. This is given by the following stochastic differential equation,
    \begin{equation}\label{eq:stochastic}
        dA_t = Y_0(A_t) dt + Y_1(A_t) dW_T, \qquad A_0 = (\omega',\nu'),
    \end{equation}
    where $\{W_t\}_{t\ge 0}$ is a Wiener process.
    The generator of $\{A_t\}_{t\ge 0}$ is the second order differential operator
    \begin{equation}
        \mathcal L = Y_0 +(Y_1)^2 =\nu \partial_\omega + b\partial_\nu^2.
    \end{equation}
    In this formula, the vector fields $Y_0$ and $Y_1$ are interpreted as first-order differential operators.
    Moreover, we added a scaling parameter $b>0$, modelling the relative strength of the two terms.

    It is natural to model the influence $k_\delta(\omega,\nu\|\omega',\nu')$ of neuron $(\omega',\nu' )$ on neuron $(\omega,\nu)$ at time $\delta>0$ as the transition density of the process $\{A_t\}_{t\ge0}$. It is well-known that such transition density is obtained by computing the integral kernel at time $\delta$ of the Fokker-Planck equation corresponding to \eqref{eq:stochastic}, that reads
    \begin{equation}
        \label{eq:kolmo}
	\partial_t I = \mathcal L^* I ,
	\quad\text{where}\quad
	\mathcal L^* =- Y_0+(Y_1)^2 = -\nu \partial_\omega + b\partial_\nu^2.
    \end{equation}

    The existence of an integral kernel for \eqref{eq:kolmo} is a consequence of the hypoellipticity\footnote{That is, if $f$ is a distribution defined on an open set $\Omega$ and such that $(\partial_t-\mathcal L^*)f\in C^\infty(\Omega)$, then $f\in C^\infty(\Omega)$.} of $(\partial_t - \mathcal L^*)$. The explicit expression of $k_\delta$ is well-known and we recall it in the following result, proved in Appendix~\ref{a:kernel}.

\begin{proposition}\label{p:expl-kernel}
	The integral kernel of equation \eqref{eq:kolmo} is
	\begin{equation}\label{eq:kernel}
	k_\delta(\omega,\nu\|\omega',\nu')=\frac{\sqrt{3}}{2 \pi b  \delta^2} \exp \left(-\frac{g_\delta(\omega,\nu\|\omega',\nu')}{
   b \delta^3}\right),
\end{equation}
where,
\begin{equation}
   g_\delta(\omega,\nu\|\omega',\nu') = 3
   (\omega-\omega')^2- 3
   \delta (\omega-\omega')
   (\nu+\nu')+ \delta^2
   \left(\nu^2+\nu
   \nu'+\nu'^2\right).
\end{equation}
\end{proposition}

\section{Numerical implementation}\label{sec:numerics}

For the numerical experiments, we chose to implement the proposed algorithm in Julia \cite{bezanson2017julia}. As already presented, this process consists in a preprocessing phase, in which we build an input function $I$ on the 3D contact space, a main part, where $I$ is used as the input of the Wilson-Cowan  equation \eqref{eq:WC}, and a post-processing phase, where the reconstructed sound is recovered from the result of the first part.

In the following, we present these phases separately.

\subsection{Pre-processing} The input sound $s$ is lifted to a time-frequency representation $S$ via a classical implementation of STFT, i.e., by performing FFTs of a windowed discretised input.
In the proposed implementation we chose to use a standard Hann  window (see, e.g., \cite{Press2007}):
\begin{equation}
    W(x) =
	\begin{cases}
		\displaystyle\frac{1+\cos(2\pi x/L)}{2}&\text{ if } |x|< L/2,\\
		0 &\text{ otherwise.}
	\end{cases}
\end{equation}
The resulting time-frequency signal is then lifted to the contact space through an approximate computation of the gradient $\nabla |S|$ and the following discretisation of \eqref{eq:lift-distr}:
\begin{equation}\label{eq:disc-chirp}
    I(\tau, \omega,\nu) =
    \begin{cases}
        S(\tau,\omega)& \text{if }\nu\partial_\omega|S|(\tau,\omega) =-\partial_\tau|S|(\tau,\omega),\\
        0 &\text{otherwise}.
    \end{cases}
\end{equation}

{
\paragraph*{Discretisation issues}

While the discretisation of the time and frequency domains is a well understood problem, dealing with the additional chirpiness variable requires some care. Indeed, even if we assume that the significant frequencies of the input sound $s$ belong to a bounded interval $\Lambda\subset \mathbb{R}$, in general the set $\{ \nu\in \mathbb{R} \mid I(\tau,\omega,\nu)\neq 0\}$ is unbounded. Indeed, one can check that as $(\tau,\omega)$ moves to a point where the countour lines of $|S|$ become vertical, the set of chirpinesses $\nu$'s such that $\nu\partial_\omega|S|(\tau,\omega) =-\partial_\tau|S|(\tau,\omega)$ will converge to $\pm \infty$.

In the numerical implementation we chose to restrict the admissible chirpinesses to a bounded interval $N\subset \mathbb{R}$. This set is chosen in a case by case fashion in order to contain the relevant slopes for the examples under consideration. Work is ongoing to automate this procedure.

}

\subsection{Processing}
Equation \eqref{eq:WC} can be solved via a standard forward Euler method. Hence, the delicate part of the numerical implementation is the computation of the interaction term.

As is clear from the explicit expression given in Proposition~\ref{p:expl-kernel}, $k_\delta$ is not a convolution kernel. That is, $k_\delta(\omega,\nu\|\omega',\nu')$ cannot be expressed as a function of $(\omega-\omega',\nu-\nu')$.
As a consequence, a priori we need to explicitly compute all values $k_\delta(\omega,\nu\|\omega',\nu')$ for  $(\omega,\nu)$ and $(\omega',\nu')$ in the considered domain. As  is customary, in order to reduce computation times, we fix a threshold $\varepsilon>0$ and for any given $(\omega,\nu)$ we compute only values for $(\omega',\nu')$ in the compact set
\begin{equation}
	\mathrm K_\delta^\varepsilon(\omega,\nu) = \{ (\omega',\nu')\mid  k_\delta(\omega,\nu\|\omega',\nu')\ge \varepsilon\}.
\end{equation}
The structure of $\mathrm K_\delta^\varepsilon(\omega,\nu)$ is given in the following, whose proof we defer to Appendix~\ref{a:kernel}.

\begin{proposition}\label{p:threshold}
For any $\varepsilon>0$ and $(\omega,\nu)\in \mathbb{R}^2$, we have that $\mathrm K_\delta^\varepsilon(\omega,\nu)$ is the set of those $(\omega',\nu')\in \mathbb{R}^2$ that satisfy
	\begin{gather}
		|\nu-\nu'|^2\le C_\varepsilon:=-4b\delta\log\left( \frac{2\pi b\tau^2}{\sqrt 3}\varepsilon\right),\\
		\left|\omega' -\omega+\frac{\delta(\nu + \nu')}{2}\right|
		\le \frac{\delta}{2\sqrt{3}}\sqrt{C_\varepsilon-|\nu-\nu'|^2}.
	\end{gather}
\end{proposition}

\begin{remark}
	It holds $C_\varepsilon\ge0$ if and only if
	\begin{equation}
		\varepsilon\le \frac{\sqrt{3}}{2\pi b\delta^2}.
	\end{equation}
	Indeed, for any $(\omega,\nu)\in \mathbb{R}^2$, the r.h.s.\ above corresponds to $\max k_\delta(\omega,\nu\|\cdot,\cdot)$, and thus $\mathrm K^\varepsilon(\omega,\nu) = \varnothing$ for larger values of $\varepsilon$.
\end{remark}

The above allows to numerically implement $k_\delta$ as a family of sparse arrays. That is, let $G\subset \Lambda\times N$ be the chosen discretisation of the significant set of frequencies and chirpinesses. Then, to $\xi = (\omega, \nu)\in G$ we associate the array $M_{\xi}:G \to \mathbb{R}$ defined by
\begin{equation}
	M_{\xi}(\xi') =
	\begin{cases}
		k_\delta(\xi\|\xi') & \: \text{if }\xi' \in \mathrm K^\varepsilon(\xi),\\
		0& \: \text{otherwise.}
	\end{cases}
\end{equation}
Therefore, up to choosing the tolerance $\varepsilon\ll 1$ sufficiently small, the interaction term in \eqref{eq:WC}, evaluated at $\xi=(\omega,\nu)\in G$, can be efficiently estimated by
\begin{equation}
	 \int_{\mathbb R^2} w(\xi\|\xi')\sigma(a(t-\delta, \xi'))\,d\xi' \approx
	 \sum_{\xi'\in K^\varepsilon(\xi)} M_\xi(\xi')a(t-\delta,\xi').
\end{equation}

\subsection{Post-processing}
Both operations in the pre-processing phase are inversible: the STFT by inverse STFT, and the lift by integration along the $\nu$ variable (that is, summation of the discretized solution). The final output signal is thus obtained by applying the inverse of the pre-processing (integration then inverse STFT) to the solution $a$ of \eqref{eq:WC}. That is, the resulting signal is given by
\begin{equation}
  \hat s(t) = \operatorname{STFT}^{-1}
\left(
\int_{-\infty}^{+\infty} a(t,\omega,\nu)\diff \nu
\right).
\end{equation}

The following guarantees that $\hat s$ is real-valued and thus correctly represents a sound signal. From the numerical point of view, this implies that we can focus on solutions of \eqref{eq:WC} in the half-space $\{\omega\geq 0\}$, which can then be extended to the whole space by mirror symmetry.

\begin{proposition}
It holds that $\hat s(t)\in \mathbb{R}$ for all $t>0$.
\end{proposition}

\begin{proof}
Let us denote
\begin{equation}
\hat S(t,\omega)=\int_{-\infty}^{+\infty} a(t,\omega,\nu)\diff \nu,
\end{equation}
so that $\hat s = \operatorname{STFT}^{-1}(\hat S)$. Moreover, for any function $f(t,\omega,\nu)$, we let $f^\star(t,\omega,\nu):=\bar f(t,-\omega,-\nu)$.

To prove the statement it is enough to show that
\begin{equation}\label{eq:star}
    a(t,\cdot,\cdot)\equiv a^\star(t,\cdot,\cdot) \qquad \forall t\ge0.
\end{equation}
This is trivially satisfied for $t\le 0$, since in this case $a(t,\cdot,\cdot)\equiv 0$.

We now claim that if \eqref{eq:star} holds on $[0,T]$ it holds on $[0,T+\delta]$, which will prove it for all $t\ge 0$.
By definition of $I$ and the fact that $S(t,-\omega)=\overline{S(t,\omega)}$, we immediately have $I \equiv I^\star$.
On the other hand, the explicit expression of $k_\delta$ in \eqref{eq:kernel} yields that
\begin{equation}
  k_\delta(-\omega,-\nu\|\omega',\nu') = k_\tau(\omega,\nu\|-\omega',-\nu').
\end{equation}
Then, for all $t\le T+\delta$, we have
\begin{equation*}
  \begin{split}
      \int_{\mathbb R^2} &k_{\delta}(-\omega,-\nu\|\omega',\nu')\sigma(a(t-\tau, \omega',\nu'))\,d\omega'\,d\nu'\\
  &=
\int_{\mathbb R^2}
k_{\delta}(\omega,\nu\|\omega'',\nu'')\sigma(a^\star(t-\tau, \omega'',\nu''))\,d\omega''\,d\nu''\\
&=
\int_{\mathbb R^2}
k_{\delta}(\omega,\nu\|\omega'',\nu'')\sigma(a(t-\tau, \omega'',\nu''))\,d\omega''\,d\nu''.
  \end{split}
\end{equation*}
A simple argument, e.g., using the variation of constants method, shows that these two facts imply the claim, and thus the statement.
\end{proof}

\section{Experiments}\label{s:experiments}

In Figure~\ref{fig:bars-wave} we present a series of experiments on simple synthetic sounds in order to exhibit some key features of our algorithm. These experiments can be reproduced via the code available at \url{https://www.github.com/dprn/WCA1}.

The first example, Figure~\ref{F:linear}, is a simple linear chirp, such that the dominating frequency  depends linearly on time  (i.e., corresponding to $\omega(t)=\mu t$ for some $\mu\in \mathbb{R}$). One observes that the processed sound presents the same feature but for a longer duration. The parameters in the experiment have been chosen to emphasize the effect of the modelling equation: the reconstruction should not present a tail that is as pronounced, however this allows to highlight the diffusive effect along the lifted slope.

The second example, Figure~\ref{F:interrupted}, corresponds to the same linear chirp as Figure~\ref{F:linear}, that has been interrupted in its middle section, creating two disjoint linear chirps. Thanks to the transport effect of the algorithm, the gap between the two chirps is bridged in the processed signal.
For this illustration, the interruption lasts about twice as long as the delay.

The third example, Figure~\ref{F:2bars}, consists of the sum of two linear chirps with different slopes. The slopes have been picked to suggest that linear continuations of the chirps should intersect. This is indeed what happens in the processed signal.
However, notice that the resulting crossing happens almost as a sum of the two chirps processed independently, with close to no interaction at the crossing. This is purely an effect of the lift procedure. The increasing chirp is (predominantly) lifted to a stratum corresponding to a positive slope, while the decreasing chirp is lifted to a negative slope stratum. De facto, their evolution under the Wilson--Cowan equation is decoupled in the 3D augmented space.

The fourth and last example, Figure~\ref{F:sin}, corresponds to a nonlinear chirp, roughly corresponding to choosing $\omega(\tau) = \sin (m \tau)$ in \eqref{eq:sin-omega}.
The construction of the model favors linearity in the evolution of perceived frequencies. We can observe how the more linear elements of the input result in more diffusion.

\begin{figure}[b]
\centering

\subfloat[
				Linear chirp. (Parameters:  $\alpha=55$, $\beta=1$, $\gamma=55$, $b=0.05$.)\label{F:linear}
				]{
					  \includegraphics[width = .475\textwidth]{test_1_bar_CI}
					  \hspace{.02\linewidth}
					  \includegraphics[width = .475\textwidth]{test_1_bar_CF}
 				}

\subfloat[
				Interrupted chirp. (Parameters:  $\alpha=55$, $\beta=1$, $\gamma=55$, $b=0.05$.) \label{F:interrupted}
				]{
					  \includegraphics[width = .475\textwidth]{test_inter_CI}
					  \hspace{.02\linewidth}
					  \includegraphics[width = .475\textwidth]{test_inter_CF}
 				}

\caption{
Experiments on synthetic sounds with varying frequencies. \emph{Left:} The STFT of the original sound. \emph{Right:} The STFT of the processed sound with delay $\delta=0.0625s$. Parameters vary for each experiment depending on the desired effect we wish to highlight. Each time, only the positive frequencies are shown: negative frequencies are recovered via the Hermitian symmetry of the Fourier transform on real signals.
}
\label{fig:bars-wave}

\end{figure}

\begin{figure}[t]
 \ContinuedFloat

\subfloat[
				Intersecting chirps. (Parameters:  $\alpha=53$, $\beta=1$, $\gamma=55$, $b=0.01$.) \label{F:2bars}
				]{
					  \includegraphics[width = .475\textwidth]{test_2_bars_CI}
					  \hspace{.02\linewidth}
					  \includegraphics[width = .475\textwidth]{test_2_bars_CF}
 				}

\subfloat[
				Nonlinear chirp. (Parameters:  $\alpha=53$, $\beta=1$, $\gamma=55$, $b=0.2$.) \label{F:sin}
				]{
					  \includegraphics[width = .475\textwidth]{test_nonlinear_CI}
					  \hspace{.02\linewidth}
					  \includegraphics[width = .475\textwidth]{test_nonlinear_CF}
 				}

\caption{Experiments on synthetic sounds with varying frequencies. (Continued.)
}

\end{figure}

\section{Conclusion}

In this work we presented a sound reconstruction framework inspired by the analogies between visual and auditory cortices.
Building upon the successful cortical inspired image reconstruction algorithms, the proposed framework lifts time-frequencies representations of signals to the 3D contact space, by adding instantaneous chirpiness information. These redundant representations are then processed via adapted diffeo-integral Wilson-Cowan equations.

The promising results obtained on simple synthetic sounds, although preliminary, suggest possible applications of this framework to the problem of degraded speech.
The next step will be to test the reconstruction ability of normal-hearing humans on originally degraded speech material compared to the same speech material after algorithm reconstruction.
Such an endeavour will contribute to the understanding of the auditory mechanisms emerging in adverse listening conditions. It will furthermore
help to deepen our knowledge on general organization principles underlying the functioning of the human auditory cortex.

\section*{Acknowledgments}
The first three authors have been supported by the ANR project SRGI ANR-15-CE40-0018
and by the ANR project Quaco ANR-17-CE40-0007-01.
This study was also supported by the IdEx Universite de Paris, ANR-18-IDEX-0001, awarded to the last author.

redpajama_set_name: RedPajamaArXiv


text:
OUC Delegation Attends Seventh International Conference on Educational Innovation through Technology (EITT 2018)
The Seventh International Conference on Educational Innovation through Technology (EITT 2018) was held from 12-15 December 2018 in Auckland, New Zealand, with the theme of "Educational Innovation through the Internet." Yuan Wei, deputy director of the Faculty of Science and Engineering at the Open University of China (OUC), together with four other OUC delegates, attended the conference.
The EITT 2018 was sponsored by the Society of International Chinese in Educational Technology (SICET), which aims to establish and strength academic and international connections, exchanges, research, and scholarship among Chinese scholars in educational technology, and advocate the application of educational technology in Chinese educational settings.
Professor Kerry Taylor, dean of the College of Humanities and Social Sciences of Massey University, delivered a welcome speech on behalf of the conference organiser. Dr. Michael Li presided over the conference. Professor Cynthia White from Massey University; Dr. Huang Yueh-Min, a chair professor from National Cheng-Kung University; Dr. Han Xibin, a professor at the Institute of Education at Tsinghua University; and Dr. Wang Yuping, an associate professor from the Griffith Institute, all gave key note speeches. Over seventy scholars and postgraduate students from six countries attended the conference and exhibited their research results in educational technology.
During the sessions, Mr. Ren Ran from the OUC Department of Information Technology gave a special speech on the research and practice of Chinese maker education and the development trends in the industry. He explored maker education development trends from the perspectives of primary education, higher education, and community education with regard to research and practice, as well as qualitative and quantitative studies. Yuan Wei, deputy director of the OUC Faculty of Science and Engineering, and Xue Haifeng, an associate professor from the OUC Department of Information Technology, attended the first and second parallel sessions and held discussions and exchanges with other attendees regarding the OUC's experience of the innovation of educational technology and the application of technology in open and distance education.
In addition, the OUC delegation introduced a series of books entitled World Open Universities Case Study Series, a key OUC "13th Five-Year Plan" project, to Professor Kerry Taylor, Professor Cynthia J White, and Dr. Li.
The OUC delegation was also invited to visit Massey University. Professor Michael Belgrave and Dr. Michael Li introduced the university's history and development, current schooling system, and teaching model. Professor Michael Belgrave also presented a gift, his book History of Massey University, to the OUC delegation.
The OUC delegation and the Massey University agreed that Professor Kerry Taylor and Dr. Micheal Li will visit the OUC in the near future. Both sides will meet again to discuss possible avenues of cooperation, including cooperation on the World Open Universities Case Study Series.
By Zheng Jipeng, OUC


Transparent page for large postcards. Capacity: 80 pages.
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Yakuza 4's PS4 remaster will be hitting Japanese stores on January 17, 2019. The news came from the latest Famitsu magazine [Thanks, Gematsu] and also revealed a few details about changes to the game. The biggest shock is that protagonist Masayoshi Tanimura has been completely recast and will be sporting a new character model. Toshiki Masuda will now assume the role in lieu of the original actor, Hiroki Narimiya.
There is actually a lot of baggage associated with Tanimura's original actor. Allegations of drug use essentially ruined his career in acting and even led him to retire altogether. He was cleared in 2016, but the damage had been done and most companies ceased associating with him. Recasting him for this remaster is likely being done to avoid royalties owed for his performance.
It's the same thing that happened with Yakuza 3's remaster on PS4. The hostesses in the game had used the likeness of real-life actresses, so they were recast to avoid any conflicts. The bigger issue is that a lot of content went missing from Yakuza 3 on PS4, which could be the case with this upcoming port as well. The cut stuff isn't that big of a deal, but it makes previous claims of no content being edited false.
That being said, there is still no confirmation of a Western release for these remasters. I would assume they are coming over, but probably as a single package instead of three releases. With these being a little more involved than simple ports, that is likely why Sega hasn't announced anything yet.


Morrison student, Cameron, had a day to remember when he took along his Japanese Spitz puppy to the world renown Crufts competition.
Morrison student, Cameron, had a day to remember when he took along his Japanese Spitz puppy to the world renown Crufts competition. Cameron, 11, and 9 month old Dottie took first place in the 'Best Puppy' category for Japanese Spitz and were awarded with a huge rosette. Cameron said " It was a fantastic day but very, very long. Dottie behaved so well and I was so proud that she got the award. Thankfully though she waited until after the judging and awards presentation before she jumped into a water bowl and got a soaking. That would have been really bad. She is a lovely fluffy puppy and is my favourite out of the four we have. Her Mum, Wren, also competed and came fourth in the veteran category. It was the best time ever!". Cameron has been an annual visitor to Crufts since he was a little boy but now takes more responsibility with the dogs so takes Dottie to training classes every Monday.


New York Times bestseller, Zillow Talk, the first publication from Zillow's CEO (Spencer Rascoff) and Chief Economist (Stan Humphries), came out in paperback early this year and reminded us how important amenities are to home value.
With data spanning from 1997 to 2014, the result was clear. Homes within one mile of Trader Joe's or Whole Foods saw a boost in value from about the time the stores opened up in the area. Two years later, homes were worth twice as much as the median value of homes throughout the U.S. That's a boost of about 10% to home value.
It's not just upscale grocery chains that boost home value. The way people search for a new home has completely shifted. They aren't just looking for a house, they are looking for a lifestyle. So what kind of amenities are making home values skyrocket?
Walkable neighborhoods are good for health, the environment, and finances. Studies find that people who live in walkable neighborhoods weigh up to 10 pounds less and are happier than those who live in traditional neighborhoods. Cars are one of the largest of household expenses and walkable neighborhoods allow many to do without, easing demand on the budget.
Studies show that activity in community events drops 10% for every 10 minutes a person spends commuting. Walkable neighborhoods get people out. They naturally meet and get involved. The public spaces in new urban developments encourage interaction within the community.
Participating in community includes meeting people, volunteering, supporting schools, buying local, and engaging with the diverse world around you—all healthy activities for individuals and the broader world. Researchers say that as technology and modern lifestyle detaches us from community, that new urbanization could potentially keep us all mentally, emotionally, and physically healthy.
Sustainability is, simply put, the ability to endure. This is an important amenity for homebuyers. A self-sustaining neighborhood keeps everything close to home which allows it to function free from dependence on the outside world. Each member of the community helps support the system.
It is also a very clean way to live. The food, transportation, and economy in these neighborhoods supports an enduring model that protects the environment within the community and throughout the world. Home designs can be an amenity in themselves if they employ the wise use of space and energy efficient appliances and materials.
The emphasis on sustainably produced items in these new urban neighborhoods can be a vehicle for a tremendous amount of change. The focus is on quality rather than quantity helps support environmentally-wise practices and the end of demand on cheap, poorly produced materials.
The homes in these neighborhoods use organic materials that eliminate unnecessary chemicals and protect the health of their residents. There is no more priceless amenity than health. Homeowners demand organic produce and goods for consumption as a part of their lifestyle.
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The lack of human capacity has been identified as a key constraining factor to sustainable forest management (SFM) in most of the diagnoses and analyses to assess progress towards SFM carried out in ITTO member countries over the past quarter century. One of ITTO's most effective contributions to addressing this constraint has been its Fellowship Program, among the Organization's earliest and most successful initiatives and probably the world's only such support program focused specifically on tropical forests and their management.
This issue of the TFU is shorter than usual due to funding constraints and is not being circulated to our 15,000 subscribers (it will only be available online, with a limited number of copies printed for the 47th ITTC session). Funding constraints have also affected the publication schedule/frequency of the TFU in 2011. It is hoped that the regular length and publication schedule of the TFU can be resumed in 2012 through the provision of sufficient voluntary contributions or through the inclusion of this work in ITTO's core activities budget under the terms of the International Tropical Timber Agreement, 2006 when it enters into force.
ITTO's Fellowship Program – can we make it even better?


Card Text: Strive — Aerial Formation costs more to cast for each target beyond the first.
We do not currently have any combos that utilize Aerial Formation. If you have your own Aerial Formation combo, we would greatly appreciate it if you would consider Adding Your Combo to our site.
We do not currently have any decks that use Aerial Formation. If you have your own deck that has Aerial Formation in it, we would greatly appreciate it if you would consider Adding Your Deck to our site.
Aerial Formation was originally printed, and so far only exists in the Journey Into Nyx magic the gathering set.


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Each day, millions of comics fans can't help but see a little bit of themselves or someone they know in Leroy and Loretta Lockhorn. The award-winning Lockhorns panel gently spoofs the state of marital bliss, poking fun at the foibles of both partners to the amusement of a worldwide audience in more than 500 newspapers. Their snappy repartee and witty banter has made them a perennial favorite.
The bickering couple's daily bouts began in 1968 with the comic collaboration of prolific cartoonist Bill Hoest and his wife, Bunny. Hoest also produced Laugh Parade and Howard Huge for Parade magazine, and the newspaper comics Agatha Crumm and What A Guy!
Since the death of her husband in 1988, Bunny Hoest has been making sure the laughter lives on with the help of her assistant, artist John Reiner.
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