Archive for the ‘Neuroscience’ Category

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!

Academic Papers as a Learning Method

Probably one of the toughest things I had to do as an undergraduate was learn how to read scholarly journal articles. Unfortunately, for those who are in the sciences, these papers are the primary means for communication of scientific questions and results. They are also an effective way for students to learn concepts and techniques common in their field. Although I claim no mastery of reading academic literature, I have learned a few tips over the last few years that I hope will be of help to any who read this. These tips will probably also apply to the other sciences.

When you want to read an article, I recommend having a pen in had to underline parts you think are especially important or that teach things you want to remember (or be able to refer to in the future). Remove all distractions. I usually listen to music when I work. When I need to read a paper, that's a bad idea. I've also found that reading aloud helps me to focus and think about what I'm reading. You can't expect this to be a speed read. It takes time, focus, and patience to succeed.

Those who, like me, are relatively new to their field, should make an effort to read entire papers. Sometimes it is tempting to just look at the figures. Although this is okay in a pinch, I have found that a single paper can help familiarize me with a lot of information if I take the time to read it from beginning to end. That being said, not all the sections are created equally, so I'll explain what I think is the relative use and importance of each of them.

  1. Start with the title - Sounds obvious, doesn't it? The title, when well written, can tell you a lot about what the paper will--and will not--contain.
  2. Read the abstract - If you're lucky, this little overview of the paper will give you all you need to know about that subject. It will almost certainly help you decide whether or not it is worth your time to keep reading.
  3. Look over the figures - Herein is the story of the paper. The figures represent the findings of the study, and, if they are properly labeled, should contain enough information to allow readers with some familiarity of the field to understand those findings.
  4. Introduction - These sections I have found particularly valuable in helping me become familiar with my field of neuronal dynamics and electrophysiology. The authors usually start with some very general statements and then narrow their focus. Introductions contain a lot of references which can be helpful if you're looking for other, related papers in your field.
  5. Discussion/Conclusions - This is similar to the introduction, except it is aimed at the future, not the past.
  6. Results - The importance of the results depends on who is reading them. As someone trying to understand the background and experiments, the results are not as important to me. As I become more comfortable with why the questions were being asked and how the results were obtained, I suspect I will be more interested in the results themselves.
  7. Methods - This is one of the most difficult parts of the paper to read. It contains a lot of chemical dosages, parameters, techniques. Here is probably where the learning curve is the steepest, and your information learned/minutes reading will be the least.

Anyhow, these are my current opinions regarding reading scientific journal articles. It is a satisfying feeling to make it through a paper and feel like you understood the majority of it. What things have you learned from your experiences reading papers?

An Overview of Voltage Clamp

Voltage clamp is an essential tool in any electrophysiology lab. It is used to look at the currents entering and exiting a cell at a set, or "clamped", voltage. First, I will explain the basic methods behind voltage clamp. Then, I will explain the uses of it.

How do I use voltage clamp?

The most important aspects of voltage clamp are shown in the diagram below, taken from here. The brown and beige sphere at the center of the stage is the cell from which we record. Finely tipped glass pipettes pierce the membrane of the cell which reseals against the surface of the glass. The metal electrodes are bathed in a solution inside the pipettes that become continuous with the cytoplasm inside the cell once the cell is punctured. The electrode on the right, the voltage electrode, records the voltage of the inside of the cell relative to the voltage outside of the cell at any given time. This signal is sent to the computer. Whenever the voltage inside of the cell deviates from a preset number (e.g. 80 mV), the computer then delivers a current back to the cell that returns the voltage back to that predefined value. The process is very quick.

By measuring the amount of current it takes to return the cell voltage back, we can measure the amount of current that is passing into or out of the cell at the set voltage. This is useful in a number of applications, some of which are explained in the next section.

What are the uses of voltage clamp?

Voltage clamp and its spin-offs have a wide range of applications in electrophysiology. As explained above, one application is the ability to record currents in and out of cells at given voltages. Resting membrane potentials can also be recorded, where the voltage is not clamped to any number. By using a similar method called patch clamp, in which a patch of the membrane is sealed onto the end of a glass pipette, scientists can even record the currents passing either direction through a single protein channel. And with even more advanced methods, such as dynamic clamp, we can simulate conductance changes in order to see how the cell would respond if the composition of its ion channels were different.

Through voltage clamp, we can elucidate a large variety of cellular mechanisms and behaviors. Some scientists use voltage clamp to look at how mutations in heart ion channels change the way a cell behaves under different voltages. Other scientists use voltage clamp to look at the behavior of neurons under simulated synaptic stimulation. How do you or would you like to use voltage clamp in your research?