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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

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Damian Reply
absolutely yes
Daniel
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it is a goid question and i want to know the answer as well
Maciej
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Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
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SUYASH Reply
for screen printed electrodes ?
SUYASH
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s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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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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