# 8.7 Mathematical modeling of hippocampal spatial memory with place  (Page 6/6)

 Page 6 / 6
${n}_{1}\ge -\frac{\tau }{I}·ln\left(1,-,\frac{{v}_{th}-{v}_{r}}{{w}_{inp}},·,\left(1,-,{e}^{-I/\tau }\right)\right).$

In the computational method, the Matlab program compT.m calculates ${n}_{1}$ by updating $v\left(t\right)$ with equations [link] and [link] . In the analytic method, AnalysisT.m calculates ${n}_{1}$ with equation [link] .

[link] shows that ${n}_{1}$ is clearly a step-wise function where higher input weights allow the cell to spike after fewer input spikes.

## Application of findings

We apply the results from the two previous sections to solve a couple simple questions. Consider the situation depicted in [link] : the rat is in Place Field 1, Cell 1 receives input with weight ${w}_{ext}$ from an external source, and, when Cell 1 spikes, it gives internal input to Cell 2 with weight ${w}_{12}$ . We can find the minimum weight of ${w}_{12}$ necessary for Cell 2 to spike in Place Field 1, or where Place Field 1 and Place Field 2 overlap.

For a fixed interspike interval $I$ of external input of fixed weight ${w}_{ext}$ , we calculate ${n}_{1}$ using equation [link] as a function of $I$ and ${w}_{ext}$ . Thus ${n}_{1}$ denotes the minimum number of external input spikes of weight ${w}_{ext}$ necessary for Cell 1 to fire. Let us denote the time of Cell 1's first spike as

${t}_{1}=\left({n}_{1}-1\right)I.$

While the rat is in Place Field 1, Cell 2 only receives input from Cell 1. To find the interspike interval of Cell 1's spikes, or equivalently the interspike interval that Cell 2 receives input in Place Field 1, consider [link] . We know that Cell 1's first spike is at $t={t}_{1}$ . After it spikes, the voltage decays until the next input spike at ${n}_{1}I$ . Then it takes another $\left({n}_{1}-1\right)I\phantom{\rule{4pt}{0ex}}ms$ for Cell 1 to spike at $t={n}_{1}I+\left({n}_{1}-1\right)I$ . Thus we subtract the first spike time from the second spike time

${n}_{1}I+\left({n}_{1}-1\right)I-\left({n}_{1}-1\right)I={n}_{1}I,$

and we see that after the first spike time at ${t}_{1}=\left({n}_{1}-1\right)I\phantom{\rule{4pt}{0ex}}ms$ , Cell 1 fires every ${n}_{1}I\phantom{\rule{4pt}{0ex}}ms$ .

Thus, we know that Cell 1 gives input to Cell 2 with weight ${w}_{12}$ every ${n}_{1}I\phantom{\rule{4pt}{0ex}}ms$ . To find the minimum weight of ${w}_{12}$ that would allow for Cell 2 to fire in Place Field 1, we consider two cases where the first is simpler: Place Field 1 is infinitely long or finitely long.

Equation [link] tells us that if

${w}_{12}\ge \left({v}_{th}-{v}_{r}\right)\left(1-{e}^{-{n}_{1}I/\tau }\right),$

then Cell 2 would fire given a sufficiently long Place Field 1. Thus, if Place Field 1 is infinitely long, we simply require that equation [link] be true, then Cell 2 fires in Place Field 1. As we do in "Number of input spikes versus input weight" , we choose an interspike interval: let ${n}_{1}I=40\phantom{\rule{4pt}{0ex}}ms$ . For this value of ${n}_{1}I$ , App.m calculates the minimum of equation [link] to be $13.83\phantom{\rule{4pt}{0ex}}mV$ , as marked in [link] .

Now assume Place Field 1 is finitely long. We apply equation [link] to find ${n}_{2}$ , where ${n}_{2}$ is the minimum number of input spikes of weight ${w}_{12}$ necessary for Cell 2 to fire. We let ${n}_{1}I=40\phantom{\rule{4pt}{0ex}}ms$ and consider only the values of ${w}_{12}\ge 13.83\phantom{\rule{4pt}{0ex}}mV$ as calculated for ${n}_{1}I=40\phantom{\rule{4pt}{0ex}}mV$ in equation [link] . [link] shows ${n}_{2}$ as a function of these values of ${w}_{12}$ . As in "Number of input spikes versus input weight" , ${n}_{2}$ is step-wise like ${n}_{1}$ .

The first spike of Cell 2 occurs at time ${t}_{2}=\left({n}_{2}\right)\left({n}_{1}I\right)-I$ . We find the value of ${w}_{12}$ necessary for ${t}_{2}\le$ the time spent in Place Field 1. Suppose the time spent in Place Field 1 is $50\phantom{\rule{4pt}{0ex}}ms$ . We see from [link] that

${n}_{2}=\left\{\begin{array}{cc}2\hfill & \phantom{\rule{4.pt}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}13.83\le {w}_{12}\le 15.95\hfill \\ 1\hfill & \phantom{\rule{4.pt}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}15.95<{w}_{12}\le 20.00\hfill \end{array}\right)$

and calculate that

${t}_{2}=\left\{\begin{array}{cc}60\hfill & \phantom{\rule{4.pt}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}13.83\le {w}_{12}\le 15.95\hfill \\ 20\hfill & \phantom{\rule{4.pt}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}15.95<{w}_{12}\le 20.00\hfill \end{array}\right)$

Thus, for $13.83\le {w}_{12}\le 15.95\phantom{\rule{4pt}{0ex}}mV$ , ${t}_{2}=60\phantom{\rule{4pt}{0ex}}ms$ and Cell 2 will not fire in Place Field 1, but for $16.01\le {w}_{12}\le 20.00\phantom{\rule{4pt}{0ex}}mV$ , ${t}_{2}=20\phantom{\rule{4pt}{0ex}}ms$ and Cell 2 will fire in Place Field 1. The minimum value of ${w}_{12}$ for ${t}_{2}\le 50\phantom{\rule{4pt}{0ex}}ms$ is marked on [link] ( App.m ).

## Future work

Our goal is to better understand the relation between input weights and backward shift of place fields. We modeled the Double Rotation Experiment using a simple 120-cell ring and calculated the backward shift of the place fields as a function of input weights of that model [link] , [link] . We have computationally and analytically found the minimum input weight necessary for activity as a function of the interspike interval as well as the minimum number of input spikes necessary for activity as a function of input weight.We applied our findings analytically to find the minimum internal input weight necessary for place fields to overlap in a simple 2-cell model.

Future work may include constructing a code to compare our analytical results from "Application of findings" for the infinite and finite place field cases that could give us the minimum internal input weight necessary for place fields to overlap. We could also couple the equations regarding input weight changes by spike timing-dependent plasticity with the equation that gives us the minimum number of input spikes necessary for activity that is dependent upon input weights. Since the time of the first spike of a cell can be given in terms of the minimum number of input spikes necessary for activity as a function of input weight, we may be able to find a value that the time approaches. Ultimately, given a set maximum for the input weight, we would like to be able to predict the amount of backward shift of a place field using spike timing-dependent plasticity.

#### Questions & Answers

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
?
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
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
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
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.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'The art of the pfug' conversation and receive update notifications?

 By Lakeima Roberts By By