**Comp Neuroscience**

In this tutorial, we will explore the role of different currents on
the firing properties of cortical neurons. It is based on a tutorial
from the book of McCormick and Huguenard which is a DOS-based
implementation of their model for thalamic and cortical neurons. The
two papers that form the basis for this tutorial are in
*J. Neurophysiology* **68**:1373-1400 (1992). I have implemented
their model in an xpp file which is given below and will also show you
the equations. There are 11 conductances that are used. By changing
them, you can look at different currents, turning them on and off as
you'd like. The conductances are

**gNa**- sodium (microsiemens)**gNap**- persistent sodium**gK**- delayed rectifier**gK2**- slow potassium**pT**- permeability for T-type calcium (10-6*cm*/sec)^{3}**pL**- permeability for L-type calcium (10-6*cm*/sec)^{3}**gC**- fast calcium/voltage depoendent potassium**gA**- A-current**gH**- Sag current**gM**- slow muscarinic potassium current**gAHP**- slow calcium-dependent potassium**gKleak**- potassium leak**gNaleak**- sodium leak

Here is the file

# the McCormick-Huguenard channel models -- Mix and match as you like # # UNITS: millivolts, milliseconds, nanofarads, nanoamps, microsiemens # moles # cell is 29000 micron^2 in area so capacitance is in nanofarads # all conductances are in microsiemens and current is in nanofarads. # par I=0,c=.29 v'=(I -ina-ik-ileak-ik2-inap-it-iahp-im-ia-ic-il-ih+istep(t))/c # the current is a step function with amplitude ip istep(t)=ip*heav(t-t_on)*heav(t_off-t) par ip=0.0,t_on=100,t_off=200 # passive leaks par gkleak=.007,gnaleak=.00265 Ileak=gkleak*(v-ek)+gnaleak*(v-ena) # aux i_leak=ileak # INA par gna=0,Ena=45 Ina=gna*(v-ena)*mna^3*hna amna=.091*(v+38)/(1-exp(-(v+38)/5)) bmna=-.062*(v+38)/(1-exp((v+38)/5)) ahna=.016*exp((-55-v)/15) bhna=2.07/(1+exp((17-v)/21)) mna'=amna*(1-mna)-bmna*mna hna'=ahna*(1-hna)-bhna*hna # aux i_na=ina # Delayed rectifier IK par gk=0,Ek=-105 Ik=gk*(v-ek)*nk^4 ank=.01*(-45-v)/(exp((-45-v)/5)-1) bnk=.17*exp((-50-v)/40) nk'=ank*(1-nk)-bnk*nk # aux i_k=ik # INap same tau as Na but diff activation par gnap=0 inap=gnap*map^3*(v-ena) map'=(1/(1+exp((-49-v)/5))-map)/(amna+bmna) # aux i_nap=inap # ia A-type inactivating potassium current # ia=ga*(v-ek)*(.6*ha1*ma1^4+.4*ha2*ma2^4) mainf1=1/(1+exp(-(v+60)/8.5)) mainf2=1/(1+exp(-(v+36)/20)) tma=(1/(exp((v+35.82)/19.69)+exp(-(v+79.69)/12.7))+.37) ma1'=(mainf1-ma1)/tma ma2'=(mainf2-ma2)/tma hainf=1/(1+exp((v+78)/6)) tadef=1/(exp((v+46.05)/5)+exp(-(v+238.4)/37.45)) tah1=if(v<(-63))then(tadef)else(19) tah2=if(v<(-73))then(tadef)else(60) ha1'=(hainf-ha1)/tah1 ha2'=(hainf-ha2)/tah2 par ga=0 aux i_a=ia # # Ik2 slow potassium current par gk2=0,fa=.4,fb=.6 Ik2=gk2*(v-ek)*mk2*(fa*hk2a+fb*hk2b) minfk2=1/(1+exp(-(v+43)/17))^4 taumk2=1/(exp((v-80.98)/25.64)+exp(-(v+132)/17.953))+9.9 mk2'=(minfk2-mk2)/taumk2 hinfk2=1/(1+exp((v+58)/10.6)) tauhk2a=1/(exp((v-1329)/200)+exp(-(v+129.7)/7.143))+120 tauhk2b=if((v+70)<0)then(8930)else(tauhk2a) hk2a'=(hinfk2-hk2a)/tauhk2a hk2b'=(hinfk2-hk2b)/tauhk2b aux i_k2=ik2 # # IT and calcium dynamics -- transient low threshold # permeabilites in 10-6 cm^3/sec # par Cao=2e-3,temp=23.5,pt=0,camin=50e-9 number faraday=96485,rgas=8.3147,tabs0=273.15 # CFE stuff xi=v*faraday*2/(rgas*(tabs0+temp)*1000) # factor of 1000 for millivolts cfestuff=2e-3*faraday*xi*(ca-cao*exp(-xi))/(1-exp(-xi)) IT=pt*ht*mt^2*cfestuff mtinf=1/(1+exp(-(v+52)/7.4)) taumt=.44+.15/(exp((v+27)/10)+exp(-(v+102)/15)) htinf=1/(1+exp((v+80)/5)) tauht=22.7+.27/(exp((v+48)/4)+exp(-(v+407)/50)) mt'=(mtinf-mt)/taumt ht'=(htinf-ht)/tauht # il L-type noninactivating calcium current -- high threshold par pl=0 il=pl*ml^2*cfestuff aml=1.6/(1+exp(-.072*(V+5))) bml=.02*(v-1.31)/(exp((v-1.31)/5.36)-1) ml'=aml*(1-ml)-bml*ml aux i_l=il # calcium concentration par depth=.1,beta=1,area=29000 ca'=-.00518*(it+il)/(area*depth)-beta*(ca-camin) ca(0)=50e-9 aux i_t=it # ic calcium and voltage dependent fast potassium current ic=gc*(v-ek)*mc ac=250000*ca*exp(v/24) bc=.1*exp(-v/24) mc'=ac*(1-mc)-bc*mc par gc=0 aux i_c=ic # ih Sag current -- voltage inactivated inward current ih=gh*(V-eh)*y yinf=1/(1+exp((v+75)/5.5)) ty=3900/(exp(-7.68-.086*v)+exp(5.04+.0701*v)) y'=(yinf-y)/ty par gh=0,eh=-43 # im Muscarinic slow voltage gated potassium current im=gm*(v-ek)*mm mminf=1/(1+exp(-(v+35)/10)) taumm=taumm_max/(3.3*(exp((v+35)/20)+exp(-(v+35)/20))) mm'=(mminf-mm)/taumm par gm=0,taumm_max=1000 aux i_m=im # Iahp Calcium dependent potassium current Iahp=gahp*(v-ek)*mahp^2 par gahp=0,bet_ahp=.001,al_ahp=1.2e9 mahp'=al_ahp*ca*ca*(1-mahp)-bet_ahp*mahp aux i_ahp=iahp aux cfe=cfestuff # set up for 1/2 sec simulation in .5 msec increments @ total=500,dt=.5,meth=qualrk,atoler=1e-4,toler=1e-5,bound=1000 @ xhi=500,ylo=-100,yhi=50 init v=-63,hna=.39,nk=.02,mna=.008 done

It is pretty big and this is only a single compartment model! To load
this up
**click here now.**

The initial file is set up for a passive membrane. Change **ip** to
0.25 nA. Run the simulation. You will see the potential
rise. Calculate the equilibrium potential with this much
current. Calculate the time-constant of the membrane using the values
for the leak conductances and the capacitance.

Spikes!

Now set **gNa=12** and **gK=2**. Rerun the
simulation. How many spikes? What is the firing frequency of the cell?
Block the potassium current by setting **gK=0.** What happens? The
cell is ``bistable'' there are two stable states a high potential and
a low potential. What accounts for the initial drop in the potential
shortly after the stimulus is turned on?

A-Current

Set

After-hyperpolarization

Set the total simulation time to 1000 msec. Set the
stimulus,

Rebound burst with T-current

Set

Calcium oscillations

Thalamic relay cells can generate intrinsic oscillations via an
interaction between the T-current and the sag current,

There are many other experiments you can do with this complicated model.

As a final exercise, I want you to look at a phase-plane analysis of a simplifed model for the T-current. In this model, there is only the T-current and a leak current. The activation of the calcium current is made instantaneous and thus the model is just a 2 dimensional system. I have also used the linear conductance model. Here are the equations:

# T-current model init v=-94,ht=.95 par I=0,c=.29 par ip=0,t_on=50,t_off=150 v'=(I -ileak-it+istep(t))/c istep(t)=ip*heav(t-t_on)*heav(t_off-t) # the current is a step function with amplitude ip # passive leaks par ena=45,ek=-105,eca=145 par gkleak=.007,gnaleak=.0005 Ileak=gkleak*(v-ek)+gnaleak*(v-ena) # aux i_leak=ileak # # IT and calcium dynamics -- transient low threshold # permeabilites in 10-6 cm^3/sec # par gt=2 IT=gt*ht*mt^2*(v-eca) mt=1/(1+exp(-(v+52)/7.4)) htinf=1/(1+exp((v+80)/5)) tauht=22.7+.27/(exp((v+48)/4)+exp(-(v+407)/50)) ht'=(htinf-ht)/tauht @ dt=.25,total=500,xp=v,yp=ht,xlo=-100,xhi=40,ylo=-.1,yhi=1 @ nmesh=100,bounds=1000 doneIf you fire this up, you will be in the

- 1.
- How many fixed points are there? Which are stable?
- 2.
- Starting at rest, would a depolarizing or hyperpolarizing
stimulus elicit a spike? Try using
**ip**small positive and negative to verify your answer. - 3.
- Change
**gnaleak = .001**and redraw the nullclines. How many fixed points are there and what is their stability? - 4.
- Integrate the equations. What behavior do you find? (Don't
forget to set
**ip=0**. - 5.
- Increase the sodium leak to 0.003. Draw the nullclines. It might
help to change the y-axis to -.05 to .1. How could one elicit a spike
in this case? (Hint, what does changing the current,
**I**do to the nullclines. Try**I=-.25, .25**to see what happens to the fixed point. - 6.
- Set
**I=0**and try different values for the current pulse,**ip**say -.25 and .25. You should be able see a rebound calcium spike.