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This report summarizes work done as part of the Computational Neuroscience PFUG under Rice University's VIGRE program. VIGRE is a program of Vertically Integrated Grants for Research and Education in the Mathematical Sciences under the direction of the National Science Foundation. A PFUG is a group of Postdocs, Faculty, Undergraduates and Graduate students formed round the study of a common problem. This module reproduces the work G. Palm "Towards a Theory of Cell Assemblies". This work was studied in the Rice University VIGRE/REU program in the Summer of 2010. This module builds an algorithm to find cell assemblies by Palm's definition and discusses some preliminary employment of that algorithm.

Introduction

History

Models of individual neurons vary in complexity, but in general, neurons tend to behave like this:

  • A neuron is either excited or not excited.
  • When excited, a neuron will stimulate neurons to which it has an outgoing connection. Otherwise, it will not stimulate other neurons.
  • A neuron becomes excited when it receives a sufficient amount of stimulation from other neurons.

When modeling the brain, see seek to model the collective behavior of neurons. The fundamental type of collective behavior is, by the Donald Hebb model of the brain, the cell assembly.

First introduced by Hebb, the cell assembly is, "a diffuse structure comprising cells... capable of acting briefly as a closed system, delivering facilitation to other such systems and usually having a specific motor facilitation" [link] . A cell assembly is a particular arrangement of a group of neurons with certain properties. The most salient of these properties is that a certain fractional portion of the assembly will excite the entire assembly.

By Hebb's proposal, a cell assembly represents a single concept in the brain. For instance, Hebb proposes that the corner of an abstract triangle may be represented by a cell assembly [link]

Since Hebb first discussed the concept of a cell assembly, there has been some amount of biological research supporting his ideas. For instance, the work of György Buzsáki suggests that groups of cells that fire during a given time period are correlated [link] .

Definitions

Node 1 (bright green) has the neighborhood {2, 3, 4, 5} (dark blue)
A k-core for which k=3
The process of closure for k=2: in step 2, an arbitrary 3 vertices are activated. Through each subsequent step, those vertices having at least 2 neighbors in the active set are activated in turn. After step 5, there are no more vertices to activate, so the vertices highlighted in step 5. are the closure of the vertices highlighted in step 2.

Gunther Palm defined Hebb's assembly in the concrete language of graph theory [link] . In brief, Palm's discussion constructs, or depends upon, the following definitions:

  • graph: A graph G has a vertex set, V ( G ) , and a set of edges, E ( G ) V ( G ) × V ( G ) . If u , v V ( G ) and u v E ( G ) , we say that the graph G has an edge directed from the vertex u toward the vertex v . Vertices are labeled with integers by convention.
  • neighborhood: We define the neighborhood of some vertex v V ( G ) with respect to G , call it N ( v , G ) , as { u : u v E ( G ) } ( [link] ).
  • degree: The degree of a vertex v V ( G ) with respect to G , call it D ( v , G ) , is | N ( v , G ) | .
  • subgraph: A graph g is a subgraph of G iff V ( g ) V ( G ) and E ( g ) E ( G ) . Further, g is an induced subgraph of G iff g is a subgraph of G and e E ( G ) ( V ( g ) × V ( g ) ) , e E ( g ) .
  • k-core: A subgraph g of G is a k -core iff v V ( g ) , | X | k , where X = N ( v ) V ( g ) ( [link] ).
  • minimal k-core A subgraph g of G is a minimal k-core iff it has no induced subgraphs which are k-cores.
  • maximum k-core A subgraph g of G is a maximum k-core iff G contains no k-cores h for which the | V ( h ) | > | V ( g ) | .
  • activation: We say that a vertex v V ( G ) can be either active or inactive. We generally say that, initially, an arbitrary subset of vertices M V ( G ) are active and the rest inactive. We further define a map f k ( M , G ) : ( sets of vertices , graphs ) ( sets of vertices ) which performs the following operation:
    • Take a graph G , and a set of vertices, M , where M V ( G ) .
    • v V ( G ) :
      • let Y = M N ( v , G )
      • iff | Y | k , then v R
    • Return R .
    For convenience, we add a superscript f k n ( M , G ) , where f k 2 ( M , G ) = f k ( f k ( M , G ) , G ) , f k 3 ( M , G ) = f k ( f k ( f k ( M , G ) , G ) , G ) , etc.
  • closure: We say that the closure of a set of active nodes M with respect to the graph G , call it c l k ( M , G ) , is equal to f k ( M , G ) . If f k n ( M , G ) does not converge for some sufficiently large n , then the closure of M is undefined ( [link] ).
  • k-tight: A k-core T , which is an induced subgraph of G , is k-tight iff it satisfies the following condition:
    • K where K is an induced subgraph of T and a k-core:
      • c l ( V ( K ) , G ) V ( T ) , or,
      • c l ( V ( T ) V ( K ) , G ) =
  • k-assembly: An induced subgraph A of G is a k-assembly iff V ( A ) = c l ( V ( T ) , G ) where T is a k-tight induced subgraph of G .

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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