Archive for the ‘Science’ Category

Anesthetics Change the Intrinsic Excitability of Neurons

Gamma oscillations in the brain are hypothesized to be involved in consciousness (Gray, 1994). Interestingly, general anesthetics are known to change both the incidence and frequency of gamma oscillations in the hippocampus. They are also known to increase the amplitude and decay time-constant of postsynaptic inhibitory currents (Whittington et al., 1996). No causal relationship, however, has been established between these network effects and cellular effects. In an effort to begin this description, I have measured the frequency-current relationships in CA1 pyramidal cells both under control conditions and in the presence of the anesthetic propofol.

Methods:

Using the whole-cell patch-clamp method, I delivered a series of input currents to the neurons. Whole-cell patch-clamp is a method where a finely-tipped glass micropipette is filled with an artificial intracellular fluid and brought down to the surface of a cell. Negative pressure (sucking) allows the cell to form a seal with the pipette and rupture the section of membrane just under the pipette tip. The intracellular fluid of the cell becomes continuous with the artificial intracellular fluid contained in the pipette. An electrode is inserted into the back end of the pipette, which can be used to record from and manipulate the cell electrically.

Each cell received the current steps with and then without the anesthetic drug propofol.

Results:

The most significant result that I saw was a change in the gain of the neuron (as evidenced by a change in the slope of the f-I curve) in response to propofol. I recorded the firing rates of the neurons in response to increasing steps of currents (current steps are shown in Figure 1).

Figure 1. Current steps.

Firing rates were separated into initial rates (rate during the first 0.3 seconds of pulse times) and steady-state rates (rate during the last third of pulse times). For this study, I was most interested in the steady-state firing rates. First, I recorded the f-I curves from neurons first without and then with propofol added (Figure 2; n=9). The process took about 30 minutes. There was a modest change in the gain of the neurons in response to the propofol treatment.

Figure 2. Steady-state f-I curve for the control (blue) and with the addition of propofol (red).

However, as I was doing the experiments I noticed that the gain of the neuron would change over time, independent of the addition of propofol. Therefore, I wanted to eliminate the effects of time from my analysis of the effects of propofol. I again followed the same protocol as I did for Figure 1, but without adding propofol for the second f-I curve. This second recording I called a "delayed control" (Figure 3; n=10). Here I also noticed a change in gain, but in the opposite direction.

Figure 2. Steady-state f-I curve for the control (blue) and delayed control (red).

When I compared the gain of the neurons with propofol added with the delayed recordings without propofol, significant differences were seen (Figure 4; p=0.021, unpaired t-test).

Figure 4. Bar graph of changes in gain (slope) as a result of the addition of propofol.

This is indicative of a change in the intrinsic excitability of the neurons as a result of the propofol treatment. This change in intrinsic neuronal excitability may, in addition to the synaptic effects of propofol, lead to changes in network behavior and contribute to propofol-induced anesthesia.

The entire poster for this study can be seen here: Utah BME 2011 Conference Poster.

References

Gray, CM (1994). Synchronous oscillations in neuronal systems: mechanisms and functions. Journal of Computational Neuroscience, 1(1-2), 11–38.

Whittington, MA, Jefferys, JG, & Traub, RD (1996). Effects of intravenous anaesthetic agents on fast inhibitory oscillations in the rat hippocampus in vitro. British Journal of Pharmacology, 118(8), 1977–86.

Hodgkin-Huxley Neuron Model

Although integrate-and-fire neuron models show key features of neuronal spiking patterns, they are not conductance-based models. A conductance-based model would allow us to look at or predict changes in membrane voltage in response to the different types of conductances (there are many different types of ion channels present in different densities in the neuronal membrane). In 1952, Alan Lloyd Hodgkin and Andrew Huxley developed a model that was conductance based. This general type of model is widely used even today. They described the neuronal membrane by the circuit below.

The lipid bilayer is an excellent capacitor and has a capacitance (c_m) of about 10 fF/μm^2. In the original Hodgkin-Huxley model, there were only three conductances: voltage-dependent sodium current, voltage-dependent potassium current, and leak current. These are each represented in the circuit diagram as I_{Na}, I_K, and I_L, respectively. The sodium and potassium conductances vary with the voltage, but the leak conductance does not. They are all described in the following equations, which also include an added input current (I_{app}):

 

c_m\frac{dV}{dt}=I_{Na}+I_K+I_L+I_{app}

or

c_m\frac{dV}{dt}=\bar{G}_{Na}m^3h(E_{Na}-V)+\bar{G}_Kn^4(E_K-V)+G_L(E_L-V)+I_{app}

\frac{dm}{dt}=\frac{m_\infty-m}{\tau_m}

         

\frac{dh}{dt}=\frac{h_\infty-h}{\tau_h}

         

\frac{dn}{dt}=\frac{n_\infty-n}{\tau_n}

These equations make up the basis of the Hodgkin-Huxley model. The top two equations are equivalent, both representing each of the currents in the same order. E_{Na}, E_K, and E_L are the reversal potentials for the sodium, potassium, and leak currents, respectively, and are found using the Nernst Equation. \bar{G}_{Na} and \bar{G}_K are the maximal conductances of their respective ions, and G_L is the leak conductance, which remains constant the entire time. Since the actual conductance at any given point in time varies with the voltage, we add in m, h, and n as gating variables. These each range between 0 and 1. Too see how these gating variables change depending on voltage, we must add a few more equations:

x_{\infty}=\alpha_x/(\alpha_x+\beta_x)

         and          

\tau_x=1/(\alpha_x+\beta_x)

,

where x=m, h, or n.

\alpha_m=9.6401\exp(0.0578V)          \beta_m=0.1081\exp(-0.0556V)

\alpha_h=0.0027\exp(-0.0500V)          \beta_h=7.2634\exp(0.0768V)

\alpha_n=1.1709\exp(0.0461V)          \beta_n=0.0555\exp(-0.0125V)

Thus, \alpha_x and \beta_x vary with voltage. Using these equations, we have a flexible, biologically relevant model of neuronal membrane voltage changes over time. Additional conductances, or ion channels, can be included as long as their maximal conductances, gating variables, and reversal potentials have been characterized. Synaptic conductances and noise can also be included. I used a MATLAB's ODE solver with the above equations and applied step currents (all starting at t=0) to produce the voltage traces shown below. If0 is the current step required to produce repeated spiking. Happy modeling!

P.S. Do you have a favorite software package for neuronal modeling?

Integrate and Fire Neuron Model

Integrate and fire neuron models are commonly used by computational neuroscientists because they can describe key characteristics of neuronal behavior while requiring relatively little computational power. The basic idea is that membrane voltage steadily increases as the neuron is injected with current until it reaches a threshold value, after which there is a "spike", or action potential. Then the neuron becomes refractory at a specified refractory potential for a specified amount of time. The voltage then continues to rise from the refractory potential with injected current, and so on. The voltage rise is described by the equation

V_{m(k+1)}=V^\infty_k+(V_k-V^\infty_k)e^{-\Delta{}t/\tau},

where   V^\infty{}\equiv{}V_{rest}+IR   and   \tau=RC,

In these equations, [math]V_m[/math] is the membrane voltage, [math]V_{rest}[/math] is the resting membrane potential (found using the Goldman-Hodgkin-Katz equation), [math]I[/math] is the input current, [math]R[/math] is the resistance, and [math]C[/math] is the capacitance. I implemented this model using MATLAB, and I post the code, as well as an example voltage trace, below. This model can be improved upon, and I will hopefully include the option of a relative refractory period soon. Hopefully it will be of help to those who may be looking for a start for their own models. Have any code of your own you'd like to share? Please feel free to post excerpts or links in the comments section!

Integrate and Fire Neuron MATLAB Code

Here's a PDF of the voltage trace (after some beautifying) elicited by the command "plot(-10:0.01:100,expIF_neuron(-10:0.01:100,[-60 -50 -70 40 -60],[1 1 10],2,[0 11]))" : IF Neuron Voltage Trace. Enjoy!

Rational Towers of Babel

Previously I wrote a little about my opinion on the compatibility of my scientific interests and my religious convictions (see Science vs. Religion). For the sake of clarification and because I just wanted to share some of the things I've found, I'm revisiting the topic.

I do not claim that spiritual knowledge can be proven through any scientific methods we now possess. This is illustrated in the New Testament, in 1 Corinthians 2:14: "But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know them, because they are spiritually discerned." In Science vs. Religion, it was my purpose to show that my understanding of the physical world can strengthen and enrich my testimony of God and the truths He has restored in recent times, but that understanding can never make a suitable foundation for such a testimony.

Genesis 11:4 states, "And they said, Go to, let us build us a city and a tower, whose top may reach unto heaven..." These people thought that they could reach God by building a tall tower, and that wicked notion resulted in the confounding of their languages. As ridiculous as the idea is that God may be reached through physical means, there are many today who aim to do the exact same thing, through postulates and theorems. As the late Truman G. Madsen put it in one of his lectures, "They have built a rational Tower of Babel, from which they comfort themselves with, 'We haven't heard from God, but he must still be there.'" I would also add that those who climb to the top of the tower often adopt the opposite point of view, this time saying, "We haven't heard from God, so he must not be there."

It is frustrating to see believers and non-believers alike arguing about the existence or character of God on the basis of tangible evidences. If God is real, what makes anyone think that they can possibly discover Him in this way, especially when it is directly contrary to the manner He prescribes? Faith is the formula for conviction.

A belated merry Christmas and a happy New Year to all!

Note: Perhaps in a later post I will describe some of the specific reasons I believe in God, as well as reasons for my equally strong conviction that the Church of Jesus Christ of Latter-day Saints (Mormonism) is true. Until then, here's the link to my page on Mormon.org where I describe some of my beliefs.