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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

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
what school?
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
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Introduction about quantum dots in nanotechnology
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hi
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what does nano mean?
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nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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