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
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,
$V\left(G\right),$ and a set of edges,
$E\left(G\right)\subseteq V\left(G\right)\times V\left(G\right).$ If
$u,v\in V\left(G\right)$ and
$uv\in E\left(G\right),$ 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
$v\in V\left(G\right)$ with respect to
$G,$ call it
$N(v,G),$ as
$\{u:\exists uv\in E(G\left)\right\}$ (
[link] ).
degree: The degree of a vertex
$v\in V\left(G\right)$ with respect to
$G,$ call it
$D(v,G),$ is
$\left|N\right(v,G\left)\right|.$
subgraph: A graph
$g$ is a subgraph of
$G$ iff
$V\left(g\right)\subseteq V\left(G\right)$ and
$E\left(g\right)\subseteq E\left(G\right).$ Further,
$g$ is an induced subgraph of
$G$ iff
$g$ is a subgraph of
$G$ and
$\forall e\in E\left(G\right)\cap \left(V\right(g)\times V(g\left)\right),\phantom{\rule{0.277778em}{0ex}}e\in E\left(g\right).$
k-core: A subgraph
$g$ of
$G$ is a
$k$ -core iff
$\forall v\in V\left(g\right),\left|X\right|\ge k,$ where
$X=N\left(v\right)\cap V\left(g\right)$ (
[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
$\left|V\right(h\left)\right|>\left|V\right(g\left)\right|.$
activation: We say that a vertex
$v\in V\left(G\right)$ can be either active or inactive. We generally say that, initially, an arbitrary subset of vertices
$M\subseteq V\left(G\right)$ are active and the rest inactive. We further define a map
${f}_{k}(M,G):(\text{sets}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{vertices},\text{graphs})\to \left(\text{sets}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{vertices}\right)$ which performs the following operation:
Take a graph
$G,$ and a set of vertices,
$M,$ where
$M\subseteq V\left(G\right).$
$\forall v\in V\left(G\right)$ :
let
$Y=M\cap N(v,G)$
iff
$\left|Y\right|\ge k,$ then
$v\in R$
Return
$R.$
For convenience, we add a superscript
${f}_{k}^{n}(M,G),$ where
${f}_{k}^{2}(M,G)={f}_{k}({f}_{k}(M,G),G),\phantom{\rule{0.277778em}{0ex}}{f}_{k}^{3}(M,G)={f}_{k}({f}_{k}({f}_{k}(M,G),G),G),$ 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
${f}_{k}^{\infty}(M,G).$ 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:
$\forall K$ where
$K$ is an induced subgraph of
$T$ and a k-core:
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
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?
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
how did you get the value of 2000N.What calculations are needed to arrive at it