modeling)physiological)systems)and)laboratory) fischell...

BIOE 340 Fall, 2012

Modeling Physiological Systems and Laboratory

Fischell Department of Bioengineering University of Maryland

College Park, MD

Lab 5: Ordinary Differential Equations Laboratory Assignment 5B

1. Introduction The ability to transmit electrical signals throughout the body is vital to the coordination of almost all physiological processes. The generation of a thought, the contraction of a muscle, and the initiation of a heart beat are all initiated by bioelectrical events. Each bioelectric transmission begins with the stimulation of a single cell, which is then propagated accordingly. This cellular stimulation is known as an action potential and is the basic component of bioelectricity. In this laboratory you will develop a computer simulation based upon the Hodgkin-‐Huxley model to gain a better understanding of membrane conductance and how it relates to bioelectrical stimulation (action potential), and transmission. At rest, most cells have a negative potential with respect to the surrounding extracellular fluid. This resting potential is due to the careful distribution of sodium (Na+), potassium (K+), chloride (Cl-‐), and calcium (Ca++) ions. The typical resting intracellular and extracellular concentrations of each ion may be found below in Table 1. The cell has specialized pumps in order to maintain a specific concentration gradient for each ion. In fact, a majority of the cell’s energy is directed towards fueling these pumps, which demonstrates the importance of maintaining these specific ionic gradients. It is vital for the cell to maintain specific ion concentrations, and corresponding resting potential, for the initiation of an electrical stimulation. It is through the manipulation of these ionic concentrations that the cell is able to produce an electrical stimulation to generate electrical transmission. Table 1. Comparison of ionic concentrations inside and outside a typical mammalian cell [1]

COMPONENT

INTRACELLULAR CONCENTRATION

(mM)

EXTRACELLULAR CONCENTRATION

(mM) Na+ 5-‐15 145 K+ 140 5

Mg2+ 0.5 1-‐2 Ca2+ 10-‐4 1-‐2

H+ 7 × 10-‐5 (10-‐7.2 M or pH 7.2) 4 × 10-‐5 (10-‐7.4 M or pH 7.4) Cl-‐ 5-‐15 110

The main components of an action potential include the resting phase, depolarization phase, repolarization phase, hyperpolarization phase, and then a return to resting phase. As stated previously, the resting phase potential is determined by the resting ionic concentrations in and around the cell (Table 1). Depolarization of the membrane occurs when there is an influx of positive Na+ ions into the cell. At the peak of depolarization Na+ channels become deactivated and K+ channels open, leading to the efflux of K+ from the cell. The efflux of K+ from the cell results in the repolarization of the cell. The hyperpolarization of the cell is caused by an excess of K+ leaving the cell. The cell returns to resting potential when the ion concentrations return to their resting state, in large part due to the activity of specialized pumps such as the sodium-‐potassium ATPase. A diagram of the action potential series may be found below in Figure 1.

BIOE 340 Fall, 2012



College Park, MD


-70

0

40

30

20

10

-10

-20

-30

-40

-50

-60

Mem

bran

e P

oten

tial (

mV

)

Na+ channels open and Na+ begins to enter cell

More Na+ channels open, depolarizing inside of the cell

Na+ channels close, K+ channels open, causing K+ to leave cell

K+ continues to leave cell, causing membrane potential to return to resting level

K+ channels begin to close, Na+ channels rest

K+ channels delay closing, causing undershoot

Time

1

2

3

4

5

6excitationthreshold

-70

0

40

30

20

10

-10

-20

-30

-40

-50

-60

Mem

bran

e P

oten

tial (

mV

)

Na+ channels open and Na+ begins to enter cell

More Na+ channels open, depolarizing inside of the cell

Na+ channels close, K+ channels open, causing K+ to leave cell

K+ continues to leave cell, causing membrane potential to return to resting level

K+ channels begin to close, Na+ channels rest

K+ channels delay closing, causing undershoot

Time

1

2

3

4

5

6excitationthreshold

Figure 1. Schematic of a typical action potential [2]. Electrostatic attraction and concentration gradient are the two main driving forces for the flux of ions across the cellular membrane. The two are combined into a single term, known as the electrochemical potential of an ion. The electrostatic driving force of the ion flux is derived from the separation of charge set up by the phospholipid bilayer of the membrane. In this sense, the membrane acts as a capacitor separating the negative charge on the inner side of the membrane from the positive charge on the exterior. Thus when the sodium channels are open, Na+ is drawn into the cell, attracted to the negative interior. When potassium channels are open at step 3, K+ is drawn out of the cell by its chemical gradient. The chemical driving force, concentration gradient, is based upon simple diffusion theory suggesting that transport of a species will flow from high concentration to low concentration. The cell returns to its resting concentration through use of the ATP driven sodium potassium pump.

The Nernst equation (1) is employed to quantitatively characterize the electrochemical potential of each ion:

in

outi C

CzFRTE

][][

ln= (1)

BIOE 340 Fall, 2012



College Park, MD


where R is the ideal gas constant (8.31451 j/molK), T is the absolute temperature (K), F is Faraday’s constant (96,485.3 C/mol), and z is the valence of the ion. The Nernst equation yields a Nernst potential, electrochemical potential, for each particular ion which is translated into a quantitative driving force by subtracting this potential from the membrane potential. The driving force for the transport of the ion is proportional to the absolute value of the difference between the Nernst potential and the membrane potential. In other words, when the Nernst potential equals that of the membrane potential there is no driving force for the spontaneous flux of that particular ion. On the other hand, when the difference between the Nernst potential and the membrane potential is large there is a large driving force for the transport of that particular ion. As will be discussed later, this voltage difference is used to characterize the current and fluxes of each respective ion. 2. Hodgkin-‐Huxley Model: Modeling Ionic Transport and Action Potentials The Hodgkin-‐Huxley model will be used to simulate cellular electric stimulation and action potentials. The model is based upon a relatively simple electronic circuit with four branch currents; three representing ion currents (Na+, K+, Leak), and one representing the capacitive current (See Figure 2 below).

Figure 2. Electronic diagram of the Hodgkin-‐Huxley model. Vm is the membrane voltage. Cm is the membrane capacitance. GNa, GK, GL are the respective ion conductances. ENa, EK, EL, are the respective Nernst potentials. The specific coordination of ion fluxes across the cellular membrane is the basis of an action potential. Therefore, our simulation must begin with the characterization of ionic transport. Ionic transport will be modeled after simple electrical circuit theory, more specifically Ohm’s law. As shown below, Ohm’s law linearly relates current to the voltage applied, and the resistance seen by charge.

gVI = (2)

Cm

ENa

GNa

INa

EK

GK

IK

EL

GL

IL

outside

insideVm

Cm

ENa

GNa

INa

EK

GK

IK

EL

GL

IL

outside

insideVm

BIOE 340 Fall, 2012



College Park, MD


Accordingly, each ion can be rewritten to relate the current across the membrane to its specific conductance and voltage (driving force) shown below (Equations 3 and 4). The currents may be modeled as a static current (3) or a dynamic current (4) )( ii EVgI −= (3)

))(()( ii EVtgtI −= (4) where V = Membrane Potential/Voltage. Using the equation above, it is possible to fully characterize the current-‐voltage relationship within a cell based upon the Hodgkin-‐Huxley model shown above in Figure 3 (5).

appLLKKNaNam IEvgEvgEvgdtdvC +−−−−−−= )()()( (5)

restVVv −= Both the Na+ and K+ conductance are dependent on time and voltage; therefore further modeling must be completed in order to properly characterize their behavior. The following sections go into detail about the specific modeling of K+ and Na+ conductance. 2.1 Potassium Model In order to appropriately fit the data obtained from their experiments Hodgkin and Huxley set the

maximum K+ conductance equal to Kg . The maximum conductance is multiplied by a proportionately constant n (0< n <1) which is the probability of finding any one of the four “ n ” gates of the channel in the open state (6). Thus the probability of the channel being open at any given time is equal to 4n . ),(),( 4 vtngvtg KK = (6) Assuming that n obeys first order kinetics one is able to derive a first order differential equation describing n as a function of time (equation 7).

nvnvdtvtdn

nn )()1)((),(βα −−= (7)

In the equation above nα is the rate constant of a channel particle moving from a closed state to an

open state, and mβ is the rate constant of a channel particle moving from an open state to a closed state.

Since the rate constants nα and mβ depend only on membrane voltage, which is held constant under the Hodgkin-‐Huxley voltage clamp simulation, it is possible to solve this differential equation to obtain a dynamic relationship for n (equation 8).

n

t

ennntn τ−

°∞∞ −−= )()( (8) 1)( −+= nnn βατ (9)

1)( −

∞ += nnnn βαα (10) Through curve fitting Hodgkin and Huxley were able to numerically describe nα and nβ (11, 12).

BIOE 340 Fall, 2012



College Park, MD


1)

1010exp(

)10(01.

−−−

= vv

nα (11)

)80

exp(125. vn

−=β (12)

2.2 Na+ Model For Na+ ionic currents, the overall modeling approach is the same as with K+. The maximum Na+ conductance is defined as Nag , m (0<m <1) is defined as the activation parameter, and h is defined as the inactivation parameter ((0< h <1). Therefore, the probability that a Na+ channel is open is approximately equal to hm3 . The equation for the dynamic conductance of Na+ may be found below (13). ),(),(),( 3 vthvtmgvtg NaNa = (13)

Similarly to K+ conductance m and h can be assumed to obey first order kinetics (14, 15).

mm

dtdm

mm βα −−= )1( (14)

hh

dtdh

hh βα −−= )1( (15)

These first order differential equations may be solved under voltage clamp conditions to determine dynamic first order equations for m and t (16, 17).

m

t

emmmtm τ−

°∞∞ −−= )()( (16)

h

t

ehhhth τ−

°∞∞ −−= )()( (17) where

mm

mmβα

α+

=∞

mmm βατ

+=

1

hh

hhβα

α+

=∞

hhh βατ

+=

1

From experimental observations one is able to simplify the equation for Na+ conductance. Since m is an increasing function it may assumed that there is total inactivation of the channel, thus ∞h is approximately equal to zero. Additionally, since Na+ conductance is negligible at rest one is able to set

°m equal to zero resulting in equation 18 below.

h

t

m eehmgvtgt

NaNa

ττ

−−

°∞ −= 33 )1(),( (18)

As with K+, Hodgkin and Huxley were able to develop analytical expressions for mα , mβ , hα , and hβ(19,20).

BIOE 340 Fall, 2012



College Park, MD


,1)

1025exp(

)25(1.

−−−

= vv

mα )18

exp(4 vm

−=β (19)

),20

exp(07. vh

−=α { } 11

10)30(exp −+⎟⎠

⎞⎜⎝

⎛ −=

vhβ (20)

3. Objectives 3.1 Experimental Analysis A working code has been provided. First, develop an m-‐file from that code and gain an understanding of the code. Add comments to each part of the code to identify the purpose of the line. Second, add a plotting regime to the m-‐file you develop in order to plot the time profiles of the potential,v , the gating variables n, m, and h, and the conductances, gk and gNa. Additionally, calculate and plot the steady-‐state values of the time constants and the gating variables as functions of the potential in the range of voltages from -‐100mV to +100mV. The code should produce 5 well-‐labeled figures when run. Include thorough comments. Include your code with comments in the appendix section of your lab report. Lastly, you will complete an experimental analysis. The objective of your experimental analysis is to investigate how certain variables affect the generation and transduction of an action potential. Develop a hypothesis, investigate the validity of that hypothesis through investigation, discuss any findings, and formulate a conclusion based upon the data obtained. A few sample “experiments” are listed below, yet novel experiments are encouraged.

• What effect does the resting potential (Vr) have on the action potential? Is there a range of Vr that must be maintained in order to elicit an action potential?

• What effect do the maximum Na+ and K+ conductance, Nag and Kg , have on an action potential? What happens when either of these conductances is set to zero?

• Investigate the influence of the Nernst potential for each ion. What happens if the Nernst potential for Na+ and K+ are switched?

• Investigate the influence of the initial gating parameters, hmn ,, . What happens when they approach zero? What happens when m exceeds h?

Due November 1: A full laboratory report summarizing the experiment conducted and the results obtained. Please submit a hard copy of the report by 3:30pm on the due date (see below for detailed description of the lab report format/requirements). Additionally, email your completed m-‐file(s) to [email protected] by 3:30 PM on Thursday, November 1st. Your code must run and produce the required figures when executed.

BIOE 340 Fall, 2012



College Park, MD


%Lab 5 %Hodgkin-Huxley Model function hhm close all v = 0; [an,bn,am,bm,ah,bh] = rate_constants(v); tn = 1./(an+bn); n_ss = an.*tn; tm = 1./(am+bm); m_ss = am.*tm; th = 1./(ah+bh); h_ss = ah.*th; sprintf('The steady state values of the gating variables are: ' ) sprintf('n_ss = %g', n_ss) sprintf('m_ss = %g', m_ss) sprintf('h_ss = %g', h_ss) vrest = -90; V = -82; v0 = V -vrest; y0 = [v0, n_ss, m_ss, h_ss]; tspan=[0,30]; [t,y] = ode45(@hodgkin_huxley_equations,tspan,y0); gbark = 36; gbarna = 120; gk = gbark*y(:,2).^4; gna = gbarna*y(:,3).^3.*y(:,4); %Here you must plot your results based on the solutions that are returned to your function. There should be 5 well-labeled plots in total as defined in the assignment. v = -100:1:100; [an,bn,am,bm,ah,bh] = rate_constants(v); tn = 1./(an+bn); n_ss = an.*tn; tm = 1./(am+bm); m_ss = am.*tm; th = 1./(ah+bh); h_ss = ah.*th;

BIOE 340 Fall, 2012



College Park, MD


function dy = hodgkin_huxley_equations(t,y) gbark = 36; gbarna = 120; gbarL = 0.3; vK = -12; vNa = 115; vL = 10.6; Iapp = 0; Cm = 1; v = y(1); n = y(2); m = y(3); h = y(4); [an,bn,am,bm,ah,bh] = rate_constants(v); dy = [(-gbark*n^4*(v-vK)-gbarna*m^3*h*(v-vNa)-gbarL*(v-vL)+Iapp)/Cm an*(1-n)-bn*n am*(1-m)-bm*m ah*(1-h)-bh*h]; end function [an,bn,am,bm,ah,bh] = rate_constants(v) an = 0.01*(10-v)./(exp((10-v)/10)-1); bn = 0.125*exp(-v/80); am = 0.1*(25-v)./(exp((25-v)/10)-1); bm = 4*exp(-v/18); ah = 0.07*exp(-v/20); bh = 1./(exp((30-v)/10)+1); end end

BIOE 340 Fall, 2012



College Park, MD


Laboratory Reports A primary goal of this course is to gain an understanding of how to write laboratory reports. These reports should be written as research papers, with the following sections:

• Title Page • Abstract • Introduction • Experimental • Results • Discussion • Conclusions • References • Tables and Figures • Appendix

A reasonable approach to writing the lab report may be the following. Write the Materials and Methods section first, perhaps during the lab period. Construct all of the figures and tables, and then write the Results section. Consider the questions the report will answer, and then write the Introduction. Next, use the Introduction and Results to guide the writing of the Discussion. Summarize everything in an Abstract, and then condense and refocus the Abstract into a Conclusions section. Below is a brief discussion of each of the sections. These are only suggestions on how research papers may be written. Other strategies may also be used, but the general principle of clarity should be upheld. The entire report should be no more than 10 pages, including approximately 6 pages of text and 2 pages of figures. This guideline does not include the Appendix. Please remember: the goal of this exercise is to construct a clearly written document that describes a question and then logically presents an answer based upon experimental results. Title Page A title page must be included. Indicate the title of the lab as well as the author’s name, affiliation, and contact information. Abstract The abstract is a single paragraph. It should be considered as an independent document, so that the abstract does not rely upon any material in the body of the report and, similarly, the body of the report does not rely upon any material in the abstract. The first sentence should clearly state the objective of the experiment. If the experiment is based upon a hypothesis, which is greatly preferred, the hypothesis should be stated and followed with statements describing its basis and evaluation. The subsequent sentences describe how the experiment was carried out. The following sentences describe, with as much precision as possible without being verbose, the results of the experiment. The final sentences describe the significance of the results and the impact of this work on the general field of study. Please remember that the abstract and the rest of the text are essentially separate documents.

BIOE 340 Fall, 2012



College Park, MD


Introduction The introduction requires that a short review of the literature pertaining to the research topic. Utilizing databases such as ISI Web of Science and PubMed, find a few papers that discuss the subject under investigation and use them as the basis for your introduction. The introduction is then best constructed as a descriptive funnel, starting with broad topics and slowly focusing on the work at hand. Perhaps three to four paragraphs are needed. The first paragraph introduces the reader to the general field of study; the second paragraph describes how an aspect of this field could be improved. The final paragraph is critical. It clearly states, most likely in the first sentence of the paragraph, what experimental question will be answered by the present study. The hypothesis is then stated. Next, briefly describe the approach that was taken to test the hypothesis. Finally, a summary sentence may be added stating how the answer of your question will contribute to the overall field of study. Experimental This section should be a straightforward description of the methods used in your study. Keep explanations brief and concise. As your methods are entirely computational, organize and present these methods appropriately: you may want to refer to computational papers in the literature for guidance of what to present and the appropriate level of detail to provide. Results The Results section presents the experimental data to the reader, and is not a place for discussion or interpretation of the data. The data should be presented in tables and figures (see below). Introduce each group of tables and figures in a separate paragraph where the overall trends and data points of particular interest are noted. Note that each table and figure in the paper must be referred to in the Results section. Be succinct. Discussion The discussion section, often the most difficult to write, should be relatively easy if the previous suggestions have been followed. First, begin with a brief paragraph that again gives an overview to the work. Then, look to your last paragraph of the introduction. If you have characterized a phenomena by studying specific effects, use the results you have presented to describe each effect in a separate paragraph. If you have presented a hypothesis, use the results you have presented to construct a logical argument that supports or rejects your hypothesis. Finally, at the end of the discussion, consider the other works in the literature that address this topic and how this work contributes to the overall field of study. Conclusions Again, first introduce the work and then briefly state the major results. Then state the major points of the discussion. Finally, end with a statement of how this work contributes to the overall field of study. References

BIOE 340 Fall, 2012



College Park, MD


Include all references that have been cited in the text. Use any generally recognized format for the citations and the References section. Software such as Endnote makes citing literature particularly easy. Tables and Figures Tables and figures can be placed in a separate section after the References section; it is not necessary to spend the formatting time needed to incorporate them into the text of the document, though you may do so. Again, clarity is the key factor, especially with the graphs. The graphs should be large, with data points and axis labels in 12-‐point font. Legends can be included within the graph or in the caption. All tables and figures need a caption. The caption should identify the table or figure in bold (i.e., Figure 3), state a brief title to the table or figure, succinctly present the significant result or interpretation that may be made from the table or figure (this may be modified from the Results or Discussion section), and finally state the number of repetitions within the experiment (i.e., n=5) as well as what the data point actually represents (i.e., the data are means and the associated error bars represent standard deviations). Appendix Please provide the MATLAB code in an appendix.

BIOE 340 Fall, 2012



College Park, MD


4. References 1. Alberts B, Johnson A, Lewis J, Raff M, Roberts K, Walter P. Molecular Biology of the Cell. Fourth Edition. Garland Science Inc. 2002. 2.. Action Potential Schematic. www.arcadia.edu (2007) 3. Barr RC, Plonsey, R. Bioelectricity: A Quantitative Approach. Second Edition.Kluwer Academic/Plenum Publishers Inc. 2000. 4. K.E. Herold. Hodgkin Huxley Model of Neuron Action Potential. Written 31 March 2005.

modeling)physiological)systems)and)laboratory) fischell...

Documents