# 8.1 Dynamics of the firing rate of single compartmental cells  (Page 2/2)

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## Implementing the single-cell model

In Shriki et al., the following equations and parameters were used:

$\frac{dx}{dt}=\frac{{x}_{\infty }-x}{{\tau }_{x}}{|}_{x=h,n,b}$
${x}_{\infty }=\frac{{\alpha }_{x}}{{\alpha }_{x}-{\beta }_{x}}{|}_{x=h,n,m}$
${\tau }_{x}=\frac{\phi }{{\alpha }_{x}+{\beta }_{x}}{|}_{x=h,n,m}.$
${\alpha }_{m}=-0.1\frac{V+30}{exp\left(-0.1\left(V+30\right)\right)-1},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\beta }_{m}=4exp\left(-\left(V+55\right)/18\right),$
${\alpha }_{h}=0.07exp\left(-\left(V+44\right)/20\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\beta }_{h}=\frac{1}{\left(exp\left(-0.1\left(V+14\right)\right)+1},$
${\alpha }_{n}=\frac{-0.01\left(V+34\right)}{\left(exp\left(-0.1\left(V+34\right)\right)-1},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\beta }_{n}=0.125exp\left(-\left(V+44\right)/80\right),$
${a}_{\infty }=\frac{1}{exp\left(-\left(V+50\right)/20\right)+1},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{b}_{\infty }=\frac{1}{exp\left(\left(V+80\right)/6\right)+1}.$

We first solve for the initial condition $V\left(0\right)$ by setting ${V}^{\text{'}}\left(t\right)=0$ in since it is generally assumed that the cell is in its steady state configuration at time $t=0$ .

Then we solve using Backwards Euler, noting that , , , use $V$ from the previous time step. The result allows us to solve using Backwards Euler, govern by the equation

${V}_{j+1}=\frac{\frac{{C}_{m}}{\Delta t}{V}_{j}+{g}_{L}{E}_{L}+{\overline{g}}_{Na}{m}_{\infty }^{3}h{E}_{Na}+{\overline{g}}_{k}{n}^{4}{E}_{k}+{\overline{g}}_{A}{a}_{\infty }^{3}b{E}_{k}}{\frac{{C}_{m}}{\Delta t}+{g}_{L}+{\overline{g}}_{Na}{m}_{\infty }^{3}h+{\overline{g}}_{k}{n}^{4}+{\overline{g}}_{A}{a}_{\infty }^{3}b}.$

In MATLAB, is written as

top = v(i) + dt/Cm*(g_L*E_L+gbar_na*m_inf^3*h(i+1)*E_na+...

gbar_k*n(i+1)^4*E_k+gbar_A*a_inf^3*b(i+1)*E_k+I_app);

bottom = 1 + dt/Cm*(g_L+...

gbar_na*m_inf^3*h(i+1)+gbar_k*n(i+1)^4+gbar_A*a_inf^3*b(i+1));

v(i+1) = top/bottom;

We say that a cell has spiked, or generated an action potential, if $V>30\phantom{\rule{4pt}{0ex}}mV.$ By varying the injective current (kept constant during each run) per simulation, we count the number of spikes in that simulation. This produces an f-I curve.

## Simulation results

Shriki et. al's Figures 1 and 2 and their respective insets were reproduced below by running the single-cell model described in section 2.2.

Figure 1 is an exact replica of Shriki's Figure 1. Thus, we confirm that the active current $A$ plays an important role in linearizing the f-I curve.

In Shriki's Figure 1 inset, there is another action potential at about 50 msec, which is not realized in Figure 2. The reason could be that Shriki did not state the assumption that the cell starts at steady state, which given the parameters, came out to be -72.73 mV. It is possible that he assumed the cell starts at 65 mV or 70 mV, common rest potentials in neuroscience. If that is the case, then the model will spike at about 50 msec.

Besides the active current, the leak conductance ${g}_{L}$ governs the threshold current ${I}_{c}$ , the smallest current for which the cell will spike. This fact is captured in Figures 3 and 4, a reproduction of Shriki's Figure 2 and its inset.

## Conclusion

By reproducing Shriki Figures 1 and 2, we have confirmed that the firing rate of an active cell depends linearly on the applied current.

## Acknowledgements

This Connexions module describes work conducted as part of Rice University's VIGRE program, supported by National Science Foundationgrant DMS–0739420. I would like to thank Dr. Steve Cox for leading our Computational Neuroscience VIGRE team. Also thanks to the entire group whose members include Jay Raol, Tony Kellems, Eva Dyer, Katherine Ward, Eric Reinelt, Mingbo Cai, Ben Leung, Ryan George, and Aneesh Mehta.

## References

• Shriki, O., Hansel D., and Sompolinsky H. (2003). Rate Models for Conductance-Based Cortical Neuronal Networks. Neural Computation 15, 1809-1841.

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