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Dynamics of the Firing Rate of Single Compartmental Cells

Module by: Yangluo Wang. E-mail the author

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Summary: This report summarizes work done as part of the Hippocampus Neuroscience PFUG under Rice University's VIGRE program. VIGRE is a program of Vertically Integrated Grants for Research and Education in the Mathematical Sciences under the direction of the National Science Foundation. A PFUG is a group of Postdocs, Faculty, Undergraduates and Graduate students formed round the study of a common problem. This module verifies the existence of a linear relationship between the firing rate of an active single-cell neuron and the injected current, as noted in ``Rate Models for Conductance-Based Cortical Neuronal Networks," by Shriki et al.

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Introduction

We seek to understand how to reduce complicated neuronal models into simplified versions that still capture essential features of the neuron. It has been observed through the detailed Hodgkin-Huxley model that the firing rate of a cell depends on current input. This module examines the dynamics of an active single-compartment cell and verifies the existence of a linear relationship between the firing rate and the injected current. It will reproduce the results found in Section 2.1 of “Rate Models for Conductance-Based Cortical Neuronal Networks," by Shriki et al [1]. In particular, it verifies Figures 1 and 2 of the Shriki paper through MATLAB simulations.

Dynamics of a Single-Compartment Cell

The Hodgkin-Huxley model of a single-compartment cell obeys

C m d V ( t ) d t = g L ( E L - V ( t ) ) - I a c t i v e ( t ) + I a p p ( t ) , C m d V ( t ) d t = g L ( E L - V ( t ) ) - I a c t i v e ( t ) + I a p p ( t ) , (1)

with the following definitions:

Table 1: Variable meanings.
Variables Description
V ( t ) V ( t ) membrane potential of the cell at time tt
C m C m membrane capacitance
g L g L leak conductance
E L E L reversal potential of the leak current
I a c t i v e ( t ) I a c t i v e ( t ) active ionic currents, varies with time
I a p p ( t ) I a p p ( t ) externally applied current.

If IappIapp were kept constant in time and sufficiently large, then the cell will fire at a rate ff. There is a simple relationship between ff and IappIapp, namely, f=F(Iapp,gL),f=F(Iapp,gL), called the f-I curve. It was noted in the Shriki paper that in many cortical neurons, the f-I curve is approximately linear if II is above threshold. Our goal is to reproduce the f-I curves shown in Figures 1 and 2 of Shriki et al. through simulation of the Hodgkin-Huxley model on a single-compartment cell.

Active Ionic Currents,

In order to generate f-I curves, we must stimulate the cell with varying current input (with the current kept constant across each simulation), and count the number of times it fired each time. But first, we need to understand the active ionic currents, an integral part of Equation 1.

There are three types of active currents, Na, K, and A. The first two currents are gated by sodium and potassium channels respectively. They obey the following equations:

I N a ( t ) = g ¯ N a m ∞ 3 h ( V ( t ) - E N a ) I N a ( t ) = g ¯ N a m ∞ 3 h ( V ( t ) - E N a ) (2)
I K ( t ) = g ¯ k n 4 ( V ( t ) - E K ) . I K ( t ) = g ¯ k n 4 ( V ( t ) - E K ) . (3)

Here, ENaENa and EKEK are the reversal potentials of Na and K respectively. The A-current is a slow current gated by a type of potassium channel. It obeys

I A ( t ) = g ¯ A a ∞ 3 b ( V ( t ) - E k ) . I A ( t ) = g ¯ A a ∞ 3 b ( V ( t ) - E k ) . (4)

The A-current is introduced to linearize the f-I curve, as we shall later see in Figure 1.

Adding these currents, we get

I a c t i v e ( t ) = I N a ( t ) + I K ( t ) + I A ( t ) . I a c t i v e ( t ) = I N a ( t ) + I K ( t ) + I A ( t ) . (5)

Implementing the Single-Cell Model

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

d x d t = x ∞ - x τ x | x = h , n , b d x d t = x ∞ - x τ x | x = h , n , b (6)
x ∞ = α x α x - β x | x = h , n , m x ∞ = α x α x - β x | x = h , n , m (7)
τ x = φ α x + β x | x = h , n , m . τ x = φ α x + β x | x = h , n , m . (8)
α m = - 0 . 1 V + 30 exp ( - 0 . 1 ( V + 30 ) ) - 1 , β m = 4 exp ( - ( V + 55 ) / 18 ) , α m = - 0 . 1 V + 30 exp ( - 0 . 1 ( V + 30 ) ) - 1 , β m = 4 exp ( - ( V + 55 ) / 18 ) , (9)
α h = 0 . 07 exp ( - ( V + 44 ) / 20 ) , β h = 1 ( exp ( - 0 . 1 ( V + 14 ) ) + 1 , α h = 0 . 07 exp ( - ( V + 44 ) / 20 ) , β h = 1 ( exp ( - 0 . 1 ( V + 14 ) ) + 1 , (10)
α n = - 0 . 01 ( V + 34 ) ( exp ( - 0 . 1 ( V + 34 ) ) - 1 , β n = 0 . 125 exp ( - ( V + 44 ) / 80 ) , α n = - 0 . 01 ( V + 34 ) ( exp ( - 0 . 1 ( V + 34 ) ) - 1 , β n = 0 . 125 exp ( - ( V + 44 ) / 80 ) , (11)
a ∞ = 1 exp ( - ( V + 50 ) / 20 ) + 1 , b ∞ = 1 exp ( ( V + 80 ) / 6 ) + 1 . a ∞ = 1 exp ( - ( V + 50 ) / 20 ) + 1 , b ∞ = 1 exp ( ( V + 80 ) / 6 ) + 1 . (12)

We first solve for the initial condition V(0)V(0) by setting V'(t)=0V'(t)=0 in Equation 1 since it is generally assumed that the cell is in its steady state configuration at time t=0t=0.

Then we solve Equation 6 using Backwards Euler, noting that Equation 9, Equation 10, Equation 11, Equation 12 use VV from the previous time step. The result allows us to solve Equation 1 using Backwards Euler, govern by the equation

V j + 1 = C m Δ t V j + g L E L + g ¯ N a m ∞ 3 h E N a + g ¯ k n 4 E k + g ¯ A a ∞ 3 b E k C m Δ t + g L + g ¯ N a m ∞ 3 h + g ¯ k n 4 + g ¯ A a ∞ 3 b . V j + 1 = C m Δ t V j + g L E L + g ¯ N a m ∞ 3 h E N a + g ¯ k n 4 E k + g ¯ A a ∞ 3 b E k C m Δ t + g L + g ¯ N a m ∞ 3 h + g ¯ k n 4 + g ¯ A a ∞ 3 b . (13)

In MATLAB, Equation 13 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>30mV.V>30mV. 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: Shriki Figure 1. f-I curves of the single-cell model. Notice the linearization of the f-I curve due to the long time constant of the A current. The units for gAgA and τAτA were mS/cm2mS/cm2 and msecmsec, respectively. Notice the kinks are merely discretization errors.
Figure 1 (2.png)
Figure 2: Inset of Shriki Figure 1.
Figure 2 (fig1inset.png)

Figure 1 is an exact replica of Shriki's Figure 1. Thus, we confirm that the active current AA 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 gLgL governs the threshold current IcIc, 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.

Figure 3: Shriki Figure 2. f-I curves given gA=20mS/cm2gA=20mS/cm2, τA=20msec,τA=20msec, and different values of gLgL.
Figure 3 (3.png)
Figure 4: Inset of Shriki Figure 2. The threshold current IcIc as a function of the leak conductance gLgL.
Figure 4 (fig2inset.png)

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 Foundation grant 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

  1. 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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