In today's lecture we discuss the integrate-and-fire (IF) model neuron, where neuronal excitation is reduced to one state variable. This is a useful model because of ease of computation and simplicity. We will use the IF model as the basis for a thalamocortical (TC) neuron which is a very interesting type of cell that shows low threshold spikes (LTS) due to T-type calcium current. The TC cell will be modeled using the integrate-and-fire-and-burst (IFB) variation of the basic IF model.

# Integrate, fire, and burst model
#
init v=-64.99
init h=0.002
v'=(iapp-gl*(v-vl)-gt*minf(v)*h*(v-vt))/c
h'=if(v>vh)then(-h/tauhminus)else((1-h)/tauhplus)
global 1 v-vtheta {v=-50}
minf(v)=heav(v-vh)

iapp=step1+step2*heav(t-ton)*heav(toff-t)
par step1=0,step2=1
par ton=50, toff=350
par vtheta=-45
par vl=-65
par gl=0.035
par c=2
par vh=-70
par tauhminus=20
par tauhplus=100
par gt=0.2
par vt=120

@ dt=0.05,total=700,maxstor=10000000
@ xlo=0,ylo=-80,xhi=700,yhi=-40
done

Here are links to the files you can load and use in XPP:

Here are the three types of behavior in the IFB model, which can be run using the three settings files you can load into XPP for the model "IFB_model.ode" (above).

"Square-wave" bursting behavior in the Morris-Lecar model is possible using the parameter set for type I dynamics, i.e., where oscillations emerge with zero frequency and then the parameter φ is raised to 0.23. These parameters for the model are listed in Figure 7.7 of the chapter. Rather than use Rinzel and Ermentrouts exact equation for the slow calcium-dependent potassium current I used:

dI/dt = ε (V₀ - V),
where &epsilon = 0.05, and V₀ = -26 mV.

So the full Morris-Lecar system is:

CV′ = I_app - g_Cam_∞(V)(V - E_Ca) - g_Kw(V - E_K) - g_L(V - E_L)
w′ = φ ([m_∞(V) - m]/Τ_m)
I′ = ε (V₀ - V).

Here is the ODE file for the bursting system with three state variables ML_SqWav_3D.ode.

(1) Simulate bursting in this model for 4000 ms, plotting membrane potential as a function of time.

(2) Use XPP-AUT to calculate the bifurcation diagram for the model using the applied current as the bifurcation parameter. Make sure you edit the file so that there are just two state variables (V,w) to make the bifurcation diagram. You can use the original version of the model, which did not include the slowly changing current described here. Construct the bifurcation diagram in two stages: first calculate the equilibrium solutions. Next "grab" the Hopf bifurcation (Auto specifies this point with "HB") and then calculate the periodic solutions.

(3) Over what range of applied current does the model exhibit multistability?

(4) Does the slow current increase or decrease during the active (spiking) phase of the rhythmic bursts?

(5) Make two other time series to demonstrate that bursting oscillations can be made slower or faster by changing ε.

(6) Explain why the last several action potentials during the active phase of each burst have more space, i.e., time, between them compared to the first several spikes at the onset of each burst.

(7) In the diagram below:

(8) In the diagram below:

(9) In the diagrams below (one is a zoomed-in version):

Sketch the phase plane at I_app = 35 pA/μm². Label all fixed points as well as stable and unstable periodic orbits. Draw the stable and unstable manifolds for the saddle point (use XPP with a small Dt to get the program to draw the manifolds for you, hint: use singular points 'Sing pts' and then 'draw invariant sets'.

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Lecture 21 Top - Cellular Biophysics and Modeling - Del Negro Homepage - Applied Science - The College of William and Mary