# 8.6 Mathematical models of hippocampal spatial memory  (Page 3/13)

 Page 3 / 13 Integrate and Fire cell Circuit Diagram. Consists of leak current, capacitive current, and excitatory postsynaptic current. In our simulation, all synapses are excitatory. Figure from Mathematics for Neuroscientists .

We model the place cell's voltage using the Conductance-based Integrate and Fire model, as pictured by the circuit diagram above. Cell voltage, or the voltage difference across the cell membrane, which we define as ${V}_{m}$ , is modified via the flow of charged ions across the cell membrane. In our integrate and fire model, we implement three parallel components through which cell voltage is adjusted. The first parallel component represents our leak current resulting from the flow of chloride ions through leaky channels. The chloride ions will travel across the membrane to reach their resting potential, which we denote as ${V}_{Cl}$ or ${V}_{rest}$ (used later in the text), of -70 mV. The flow of ions is limited by a conductance ${g}_{Cl}$ , which remains fixed at a value of 1 mS/cm ${}^{2}$ . S represents Siemens, the reciprocal of the unit of resistance $\Omega$ . When we put these terms together using Ohm's law and solve for chloride current, we get:

${I}_{Cl}={g}_{Cl}\left({V}_{m}-{V}_{Cl}\right)$

The second parallel component of the integrate and fire model consists of a capacitor, indicative of the cell membrane's ability to separate electrical charge (in the form of ions). For this component, the capacitive current, ${I}_{C}$ represents the current due to the change in transmembrane voltage. This current takes the following form:

${I}_{C}={C}_{m}\frac{d{V}_{m}}{dt}$

Here, ${C}_{m}$ denotes the membrane's capacitance, or ability to store charge. We use the value of ${C}_{m}$ = 20 $\mu$ F/cm ${}^{2}$ .

The third parallel component of our circuit involves current from synaptic inputs. This input will act as the driving force for depolarization (in this setup we do not use any inhibitory synapses). As such, we use a excitatory reversal potential of ${V}_{syn}$ = 0 mV, which will help raise the cell voltage toward the firing threshold in the presence of input. The magnitude of the input current is limited by synaptic conductance, which we denote ${g}_{syn}$ in the circuit diagram (also referred to as ${g}_{E}$ ). The synaptic conductance, unlike the leak conductance of the chloride channels, is a variable conductance (denoted by the arrow through the resistor) whose magnitude is discussed in the next section. The synaptic input current is represented by the following:

${I}_{syn}={g}_{syn}\left({V}_{m}-{V}_{syn}\right)$

Since these three currents are in parallel, we can use Kirchhoff's current law, which states that:

${I}_{Cl}+{I}_{C}+{I}_{syn}=0$

Substituting values in for the currents yields the following:

${C}_{m}\frac{dV}{dt}+{g}_{Cl}\left(V-{V}_{Cl}\right)+{g}_{syn}\left(V-{V}_{syn}\right)=0$

Note that this differential equation only applies as long as the membrane voltage remains subthreshold. Having explained the voltage dynamics of the Integrate and Fire model, we now discuss how our variable synaptic conductance is adjusted.

## Synaptic conductance

In our model, we will represent input from external stimuli or neighboring place cells as an increase in synaptic conductance. We define our synaptic weight as the degree of increase in conductance of the postsynaptic cell due to presynaptic cell firing.As such, the excitatory synaptic conductance ( ${g}_{E}$ ) of each cell takes the following form:

#### 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
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    By    By By