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Hodgkin-Huxley Model and Propagation of Action Potential

Hodgkin-Huxley Model and Propagation of Action Potential



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I'm studying Hodgkin-Huxley model of action potential, and I have some confusion.

In the well-stated HH model, we have time constants for each ion currents, described as the reciprocal of the sum of forward rate and backward rate.

In the caple theory, there is another time constant that is defined as the multiple of cell resistance and cell capacitance.

Is there any relationship between the two time constants? Do we have to think the two processes, which means the 'generation' of action potential and the 'propagation' of it separately?

Thanks.


A model for propagation of action potentials in smooth muscle

A modified Hodgkin & Huxley (1952) model for axons was used to simulate smooth muscle action potentials. The modifications were such as to match our own experimental results and published data on the passive and active behavior of smooth muscle.

A brief account of the modifications introduced to the HH model is as follows. The resting ionic conductances were obtained from the data of Casteels (1969). Chloride conductance was replaced by an ad hoc leakage conductance ( g ̄ L ) in order to obtain a resting membrane resistance of about 11 kΩcm 2 . The ionic equilibrium potentials were according to Kao & Nishiyama (1969). The rate constants m, n and h have similar form to those in axons, but their different numerical values produce action potentials that match the duration of the smooth muscle action potential (about 16 ms) at half its maximum amplitude. The effective membrane capacitance was taken as 2.5 μF/cm 2 .

The results obtained by implementing those smooth muscle parameters in the HH formulation include: (a) a membrane potential that matches the main characteristics of experimentally recorded action potentials in uterine smooth muscle and guinea-pig taenia-coli, and (b) a propagated action potential which, on a cable diameter of 5 μm (similar to the diameter of a single smooth muscle cell), has a speed of propagation within the range of the values experimentally recorded in smooth muscle. This observed velocity of propagation is not compatible with a large cable and it is concluded that “functional units” are not required to sustain propagation of action potentials in smooth muscle.


Hodgkin-Huxley model of the action potential in the the squid giant axon

In the late 1940’s and early 1950’s Alan Hodgkin and Andrew Huxley elucidated the biophysical underpinnings of nerve excitation. This tutorial walk the user through the steps necessary to reproduce and understand key aspects of their Nobel Prize-winning work using the MATLAB simulation environment. TheTutorial on CellML, OpenCOR & the Physiome Model Repository shows you how to implement the Hodgkin-Huxley model using the OpenCOR model authoring and simulation tool.

Video Tutorial

Step 1: Simulating the dynamics of the potassium conductance

Potassium conductivity $g_K(t)$ is simulated as proportional to the forth power of a gating variable, $n(t)$:
$g_K=n^4 (ar ) $
where $ar$ is the maximum conductance of potassium in the cell, occurring when $n = 1$. The dynamics of potassium conductivity are simulated to respond to changes in membrane potential via a first-order process
$dn/dt=α_n (1-n)- β_n n$
where $α_n$ and $β_n$ are the rates of opening and closing of the channel. The dependency of the opening and closing rates on voltage are modeled by functions that match the observed data:
$α_n=0.01 frac<(10-v)><10> ight)-1>$
$β_n=0.125 exp left(frac<-v> <80> ight)$
where $v$ is the membrane voltage (inside cell minus outside cell) and measured relative to the resting potential of around -60 mV. The units in the above expression are in mV. A comparison of the above equations for opening and closing rates to the original data derived from Hodgkin and Huxley’s experiments is illustrated below:

Estimated opening and closing rate constants for the potassium conductance as functions of membrane voltage are show on left. These data show that as the membrane depolarizes ($v$ becomes more positive) the opening rate increases and the closing rate decreases. At high values of $v$ the channels open. At low voltages the channels close. The model captures this important feature of the nerve cell conductivities through the above equations for $α_n$ and $β_n$ that effectively capture the data measured by Hodgkin and Huxley.

The time-dependent behavior of the potassium conductance can be simulated using a program that integrates the equation for $dn/dt$. Simulation in MATLAB requires a function that returns the computed time derivative $dn/dt$ as a function of $n(t)$ and $v$. The MATLAB function dXdT_n.m has the following syntax:

,n,v)
% FUNCTION dXdT_N

% Inputs: t - time (milliseconds)

% x - vector of state variables
% V - applied voltage (mV)
% Outputs: f - vector of time derivatives
%

This function can be integrated using MATLAB to simulate a voltage-clamp experiment. For example, we can simulate the response to a voltage-clamp experiment, with voltage set to $v = 100$ mV, and initial condition $n = 0$ with the following script:

which gives the output illustrated to the right above. (See Tutorial Enzyme Kinetics with MATLAB 2 for more on how to simulate differential equations using MATLAB.)

Hodgkin and Huxley’s full analysis of the potassium ion conductivity can be reproduced using the script HH_potassium_current.m, which includes data extracted from the original publication (A. L. Hodgkin and A. F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol., 117:500–544, 1952), and uses the function dXdT_n.m to simulate the conductivity response to a series of voltage-clamp experiments. This script generates the plot below, which may be compared to Figure 3 in Hodgkin and Huxley (J. Physiol., 117:500–544, 1952).

Step 2: Simulating the dynamics of the sodium conductance

Sodium conductivity $g_(t)$ is simulated as governed by two gating variables, $m(t)$ and $h(t)$:
$g_Na=m^3h(ar<>>) $
where $ar<>>$ is the maximum conductance of sodium in the cell, occurring when $m = 1$ and $h = 1$. As for potassium conductivity, the dynamics of sodium conductivity are simulated to respond to changes in membrane potential via first-order processes
$dm/dt=α_m (1-m)- β_m m$
$dh/dt=α_h (1-h)- β_hh$
The difference is that since there are two gating variables there are two opening rates α_m and α_h and two closing rates $β_m$ and $β_h$. The equations that capture the voltage dependency of these variables are
$α_m=0.1 frac<25-v> <10> ight) -1>$
$β_m=4 expleft( frac<-v> <18> ight)$
$α_h=0.07 exp left( frac<-v> <20> ight)$
$β_h=frac<1><10> ight)+1>$
Again, v is the membrane voltage (inside cell minus outside cell) and measured relative to the resting potential of around -60 mV. The units in the above expression are in mV. Comparison of the above equations for opening and closing rates to the original data derived from Hodgkin and Huxley’s experiments is illustrated below:

These data reveal that the m and h variables operate on substantially different time scales. Like the potassium conductance, the sodium conductance opening rate increases with increasing v. However, the rates are about 10 times higher than the rates for the potassium conductance. For this reason, the channels associated with the sodium conductance have been termed fast sodium channels.
The h gating variable shows qualitatively opposite behavior to that of the m gating variable. It varies more slowly and tends to close in response to an increase in v.

The time-dependent behavior of the sodium conductance can be simulated using a program that integrates the equations for dm⁄dt and dm⁄dt: dXdT_mh.m has the following syntax:

,x,v)
% FUNCTION dXdT_MH
% Inputs: t - time (milliseconds)
% x - vector of state variables
% V - applied voltage (mV)
%
% Outputs: f - vector of time derivatives

%

% state variables
m = x(1)
h = x(2)

% alphas and betas:
a_m = 0.1*(25-v)/(exp((25-v)/10)-1)
b_m = 4*exp(-v/18)
a_h = 0.07*exp(-v/20)
b_h = 1 ./ (exp((30-v)/10) + 1)

% Computing derivatives:
f(1,:) = a_m*(1-m) - b_m*m
f(2,:) = a_h*(1-h) - b_h*h

This function can be integrated using MATLAB to simulate a voltage-clamp experiment. For example, we can simulate the response to a voltage-clamp experiment, with voltage set to $v = 100$ mV, and initial condition $n = 0$ with the following script:

which gives the output illustrated to the right. Here we use the initial condition $m(0) = 0$, $h(0) = 1$, which means that the conductivity is initially zero. Because voltage is clamped at $v = 100$ mV, the ‘fast’ conductivity associated with the m gate rapidly increases, and the slower h gate eventually closes, resulting in the biphasic transient captured by the model.

Hodgkin and Huxley’s analysis of the sodium ion conductivity can be reproduced using the script HH_sodium_current.m, which includes data extracted from the original publication (A. L. Hodgkin and A. F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol., 117:500–544, 1952), and uses the function dXdT_mh.m to simulate the conductivity response to a series of voltage-clamp experiments. This script generates the plot below, which may be compared to Figure 6 in Hodgkin and Huxley ( J. Physiol., 117:500–544, 1952).

Note that there is a slight mis-match between data and simulation. This is because these simulations use the functions for $α_m$, $α_h$, $β_m$, and $β_h$ that are associated with the best fit to values extracted from the ensemble of axons characterized by Hodgkin and Huxley, while the data on the right represent individual recordings.

Step 3: Putting it all together and simulating the action potential

The overall model is represented by the circuit model on the right, where $V_$ and $V_K$ are the Nernst potentials for sodium and potassium. Since voltage is measured as inside potential minus outside potential, for a typical cell with a value of $V_K$ of approximately −70 mV, the membrane potential and the Nernst potential work in opposition when the $V approx −70$ mV. The sodium concentration, however, is typically higher on the outside of the cell, and $V_Na$ may be in the range of +50 mV. Thus when $V = −70$ mV, the thermodynamic driving force for Na$^+$ influx is $−(V − V_) approx +120$ mV.

In the model for the action potential, membrane voltages are expressed relative to resting potential: $v = V – V_o$, where $V$ is the absolute membrane potential (inside minus outside potential) and $V_o = -56$ mV is the resting potential. The governing equation for the circuit model is

$C_m dv/dt=-g_ (v-v_ )-g_K (v-v_K )-g_L (v-v_L )+I_$

where the membrane capacitance has the value $C_m = 1 imes 10^<-6>$ μF cm$^<-2>$. The term $g_L (v-v_L )$ represents a leak current and $I_$ is the current externally injected into the cell. The Nernst potentials for the squid giant axon prep are $v_ = V_ - V_o = 115$ mV, $v_K = V_K - V_o = -12$ mV, and $v_L = 10.6$ mV.

Combining the equations for the gating variables with the equation for membrane potential, we can write a function for the full combined model:

,x,I_app)
% FUNCTION dXdT_HH
% Inputs: t - time (milliseconds)
% x - vector of state variables
% I_app - applied current (microA cm^<-2>)
%
% Outputs: f - vector of time derivatives
%

% Resting potentials, conductivities, and capacitance:
V_Na = 115
V_K = -12
V_L = 10.6
g_Na = 120
g_K = 36
g_L = 0.3
C_m = 1e-6
% State Variables:
v = x(1)
m = x(2)
n = x(3)
h = x(4)
% alphas and betas:
a_m = 0.1*(25-v)/(exp((25-v)/10)-1)
b_m = 4*exp(-v/18)
a_h = 0.07*exp(-v/20)
b_h = 1 ./ (exp((30-v)/10) + 1)
a_n = 0.01*(10-v)./(exp((10-v)/10)-1)
b_n = 0.125*exp(-v/80)
% Computing currents:
I_Na = (m^3)*h*g_Na*(v-V_Na)
I_K = (n^4)*g_K*(v-V_K)
I_L = g_L*(v-V_L)
% Computing derivatives:
f(1) = (-I_Na - I_K - I_L + I_app)/C_m
f(2,:) = a_m*(1-m) - b_m*m
f(3) = a_n*(1-n) - b_n*n
f(4) = a_h*(1-h) - b_h*h
% Outputting the conductivities
varargout <1>= [(m^3)*h*g_Na (n^4)*g_K g_L]

The model may be simulated with the script HodHux.m to generate the following output.


2.2.3 Dynamics

A
B
C
Fig. 2.6: A . Action potential . The Hodgkin-Huxley model is stimulated by a short, but strong, current pulse between t = 1 t=1 and t = 2 t=2 ms. The time course of the membrane potential u ⁢ ( t ) u(t) for t > 2 t>2 ms shows the action potential (positive peak) followed by a relative refractory period where the potential is below the resting potential u rest u_ < m rest>(dashed line). The right panel shows an expanded view of the action potential between t = 2 t=2 and t = 5 t=5 ms. B . The dynamics of gating variables m m , h h , n n illustrate how the action potential is mediated by sodium and potassium channels. C . The sodium current I Na I_ < m Na>which depends on the variables m m and h h has a sharp peak during the upswing of an action potential. The potassium current I K I_ < m K>is controlled by the variable n n and starts with a delay compared to I Na I_ < m Na>.

In this subsection we study the dynamics of the Hodgkin-Huxley model for different types of input. Pulse input, constant input, step current input, and time-dependent input are considered in turn. These input scenarios have been chosen so as to provide an intuitive understanding of the dynamics of the Hodgkin-Huxley model.

The most important property of the Hodgkin-Huxley model is its ability to generate action potentials. In Fig. 2.6 A an action potential has been initiated by a short current pulse of 1 ms duration applied at t = 1 t=1 ms. The spike has an amplitude of nearly 100mV and a width at half maximum of about 2.5ms. After the spike, the membrane potential falls below the resting potential and returns only slowly back to its resting value of -65mV.


Solving ODEs of H&H Model using R-Package deSolve

The H&H model is mathematically complex, and has no analytical solution. Solving for the membrane action potential and the ionic currents requires integration approximated using numerical methods. Here we will use the the R-Package deSolve to solve the H&H differential equations and simulate the time evolution of the membrane potential and the dynamics of the gating variables (m), (n) and (h).
The deSolve package is an add-on package of the open source data analysis system R for the numerical treatment of systems of differential equations.
The package contains functions that solve initial value problems of a system of first-order ordinary differential equations (ODE), of partial differential equations (PDE), of differential algebraic equations (DAE), and of delay differential equations (DDE).
The functions provide an interface to the FORTRAN functions lsoda, lsodar, lsode, lsodes of the ODEPACK collection, to the FORTRAN functions dvode, zvode, daspk and radau5, and a C-implementation of solvers of the Runge-Kutta family with fixed or variable time steps.
The package contains also routines designed for solving ODEs resulting from 1-D, 2-D and 3-D partial differential equations (PDE) that have been converted to ODEs by numerical differencing.

Steps to be taken

To implement and solve the H&H differential equations in R we proceed as follows: (Examples can be located at deSolve.)

Needed R-Packages

Model specification, which consists of:

The model application, which consists of:

So let’s now start with the R-coding.


The Hodgkin Huxley Model: Analysis of Dynamic Behavior of the Action Potential in the Giant Squid Axon

Abstract: The main concern of modelling a biological neuron using any electronic circuit to create qualitive models. A nerve cell reacts to a stimulus with a voltage shift or an energy potential gap between the cell and its environment resulting in a spike in voltage. To generate action potential, different methods should be implemented. To improve the propagation of action potential, we use an accurate and efficient method i.e. Hodgkin- Huxley model.

The Hodgkin-Huxley experiment is a quantitative description of the actual movement of the neuronal membrane across ion- selective channels, and demonstrated the underpinnings of cell physiology as one of the most revolutionary studies of the 20th century and beyond. Using simple, first-order, ordinary differential equations, Hodgkin and Huxley were able to explain their time behavior using potassium (K) and sodium (Na) streams of intracellular membrane potential and currents. This was done using parameters equipped with a voltage clamp test on the giant axon of the squid. MATLAB simulates the kinetics of ionic currents, effects of alteration of the component currents, and the analysis time step.

Keywords: Hodgkin-Huxley model, cell electrophysiology, biophysics, computational biology.

We should primarily concentrate on neurons, but it is prudent to begin this analysis at the most fundamental level: the brain. In brain, a cell is an entity that is the smallest structural structure capable of operating independently. Neurons are a particular type of cell. These are trained primarily in sending electric signals to other neurons. There are three separate parts of the neuron that we ought to understand: the cell nucleus, the dendrite, and the axon. The electrical impulse that brings an instruction from the brain to, for example, the hand passes along a chain of neurons [1].

The nerve cell-to-neuron signal transfer happens at the synapse, a fluid flow region between two communicative neurons. The electrical impulse of the presynaptic neurons is translated into a chemical message in the synapse and translated again into an electronic signal in the postsynaptic neurons [2][3].

1.1 Representations of biological neuronal model:

These are the most biologically reliable ones. Different model parameters represent those biological components of the neuron. Such models illustrate how the neurons work in depth but are costly to quantify and therefore the simulations appear to be sluggish. We model the capacity of a neuron to combine inputs and fire through a threshold. The Hodgkin- Huxley model is a neuron point design. Point neuron simulations are concerned mainly with how the neurons treat input voltage to generate or not create an action potential. The Hodgkin-Huxley method is a form of accurate description of the neuron. This model is concerned with how ion motions cause shifts in the voltage of the neurons. Therefore, to understand what this model mimics, the basic knowledge of ion changes is necessary [4].

The crucial argument that Hodgkin and Huxley were able to show seemed to be that two ion forms, namely sodium (Na) and potassium ( K), would adequately clarify the electrical properties of the neurons. Outside the cell the Na+ concentration is higher than inside so these ions are driven by diffusion into the neuron. Often, when sodium ions are drawn into the cell by electrical forces as the cell is

negatively charged compared to the external cell is shown in figure 1[1].

Figure 1: Ionic movement of and

The ions also operate with two forces: diffusion and electrostatic pressure, which influence the ionic flow of the extracellular and intracellular fluid. The diffusion force pushes the ions to disperse the ion uniformly throughout the liquid, with no high or low-concentration regions. Furthermore, as the intracellular fluid contains a high concentration of +, absorption pushes these ions into the extracellular fluid. Electrostatic pressure causes the same charge ions to be repelled, whereas ions with opposite charges are attracted to each other [4].

Both cells in the body have electrical impedance or potential differences throughout inside and then outside it. Because the cell membrane differentiates inner surface from the outside, this potential variation is considered as the potential of the membrane. In scientific terms, the the cell voltage is described in equation (1)

Where is the voltage of the inner cell and is the voltage of the outer cell. It is going to change during an electrical potential. Capacitance is known as a charge kept divided by the voltage needed to hold the charge [5].

This capacitance is believed to be continuous, i.e. not time-dependent. Therefore, by taking the time derivative, we can obtain the following equation from the movement of charge across the membrane.

Here R is the universal gas constant, T is the absolute temperature of Kelvin, F is the fixed value of Faraday, and Z is the valence of the particle. Because three of these parameters are environmental parameters, one often comes across different formulas where these constants have been defined by their importance in the issue under discussion.

The Hodgkin-Huxley model is one among the biological models (Abbott & Kepler, 1990), utilizes four differential equations to live the potential of the membrane. These four differential equations model the ionic activity of the brain. Hodgkin as well as Huxley carried out experiments on the giant squid axon and found 3 categories of electrons current, i.e. sodium , potassium and leakage current consisting primarily of chlorine ions. The movement of these molecules from the cellular membranes is regulated by different ion-dependent voltage mediums for sodium and potassium ions. Certain channel forms are not explicitly described with the leak current. The semipermeable cellular membrane separates the interior of the cell and acts as a capacitor. From their observations, we were able to derive comprehensive calculations to describe the changes in the ionic current intensity [6].

Figure 2: Simplified equivalent circuit for a small section of the giant squid axon

In figure (2), the input current I(t) is transferred through the cell, either providing significant charge to the capacitance or leakage along the cell membrane mediums. A resistor represents each type of channel. The unspecific channel has a resistance to leakage , a resistance to the sodium channel

, and a resistance to potassium channel

This formula is important to evaluate the membrane potential of the Hodgkin-Huxley model for all the work to be done on this topic [1].

The electrical potential produced by the electrochemical balance of the membrane, the equilibrium potential, can b calculated by a simple formula called the Nernst equation.

concentration of the ion within the neuron varies from the one in the extracellular material by the intensive transport of the ion across the cellular membrane. Nernst potential is different from every sort of ion. The sodium , potassium and undefined leakage channel voltages are , and respectively.

The conversation of electrical charge on a membrane piece indicates that the current Ix(t) added can be separated into capacitance current I that affects the condenser C and that passes across ion mediums [6].

4 VOLTAGE-CLAMP USED BY HODGKIN AND HUXLEY

The theory of electrical circuits states that the amount of

voltage shift over the capacitor is equal to the maximum current used to charge it, or

In many of the studies conducted by Hodgkin and Huxley, an electrode was injected into the squid axon to keep the membrane at a fixed voltage. Figure 3 shows the

calculated current flow from the axon of a giant squid through a specified region of neural membrane to calculate

Where the current among three channels is,

how ions move through the neuron membrane during an action potential [8]. The voltage-clamp enabled the ionic

Vx VNa + Gk Vx Vk + GL Vx VL

currents to be registered directly, flowing through the giant axon's axonal membrane without any subsequent membrane

Whereas its leakage channel is defined by a voltage dependent conductance = 1/. Because is the overall potential around the membrane and is the reversal potential across leaky channel, the voltage at the leaky channel is .

By applying ohms rule at leaky current

The time-dependent membrane current is

A simple voltage clamp can iteratively calculate the membrane potential, and then adjust the membrane potential (voltage) by applying the required current to the target value. Its "clamps" the cell membrane at the ideal voltage constant, enabling the clamp to monitor what currents are transmitted. The voltage-clamp eliminated capacitance problems and created an isopotential membrane [8].

Where , , are reversal potentials

When this mediums are free, it transfer currents among a greater conductance , , . Even then, a few streams are being blocked. m, n and h are gating aspects that are created to design the possibility that a channel will be opened at a given time. The cumulative constants of m and h handles the channels while K gates are handled by n constant is shown in equation (8).

The efficient conductivity of the sodium channels is defined as 1/ = 3 where m specifies the initiation (opening) of the medium and h the termination(closing/blocking) of the medium. Potassium

conductivity is 1/ = 4 while n represents the initiation of the channel [6].

The change in the gating variables is based on the following differential equations:

Figure 3: Voltage clamp of a squid axon

5 HODGKIN-HUXLEY MODEL USING EULERS METHOD

The computational method of Hodgkin and Huxley was one convenient for manual computation by means of the tools at that time. The task of determining the value at a given time is an integral part of the differential equation. The difficulty with this is that the four differential equations are interrelated and therefore it is difficult to calculate the integral. Then, an approximation of the equation is used, often by the Euler method. In Euler's process, there is a parameter that is the time step dt, which, based on the values selected, which make it impossible for the model to function or make it take too long to operate. A general first-order differential equation of the form.


Interpretation as a CellML model¶

We discussed the idea and implementation of encapsulation in the previous section on the sodium channel , and here it is no different. We would like to create a model with the encapsulation structure shown in Figure 25 .

Figure 25 The relationship between the ion channels for sodium, potassium and the leakage current, and the membrane and environment components. ¶

As with other aspects of libCellML, there are several options for the model construction process. Since we already have potassium and sodium channel models available, it would make sense to be able to reuse these here. This functionality requires imports their use is demonstrated in HH Tutorial 2: Creating a model which uses imports .

Importing allows all or part of a model to be used in-situ, without needing to manually parse its CellML file and instantiate it as an additional model (as has been the procedure in HH Tutorial 1: Creating a model using the API and HH Tutorial 3: Debugging a model ).


Author Summary

In 1952, Hodgkin and Huxley described the underlying mechanism for the firing of action potentials through which information is propagated in the nervous system. Hodgkin and Huxley's model relies on the opening and closing of channels, selectively allowing ions to move across the membrane. In the original picture, the channels open independently of one another. A recent paper argues that this model is incapable of modeling a set of action potential data recorded in the cortical neurons of cats. Instead the authors suggest that to model their data it is necessary to conclude that ion channels open cooperatively, so that opening one channel increases the chance that another channel opens. We analyze the initiation of action potentials using a method from theoretical physics, the path integral. We demonstrate that deviations of the data from the predictions of the Hodgkin-Huxley model hinge on measurement of the noise strength.

Citation: Colwell LJ, Brenner MP (2009) Action Potential Initiation in the Hodgkin-Huxley Model. PLoS Comput Biol 5(1): e1000265. https://doi.org/10.1371/journal.pcbi.1000265

Editor: Karl J. Friston, University College London, United Kingdom

Received: March 26, 2008 Accepted: December 2, 2008 Published: January 16, 2009

Copyright: © 2009 Colwell, Brenner. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: This work was supported by the National Science Foundation (NSF) Division of Mathematical Sciences, Kavli Institute of Theoretical Physics, and NSF.

Competing interests: The authors have declared that no competing interests exist.


Control analysis of the action potential and its propagation in the Hodgkin-Huxley model

Thesis (MSc (Biochemistry))--University of Stellenbosch, 2010.

ENGLISH ABSTRACT: The Hodgkin-Huxley model, created in 1952, was one of the first models in computational neuroscience and remains the best studied neuronal model to date. Although many other models have a more detailed system description than the Hodgkin-Huxley model, it nonetheless gives an accurate account of various high-level neuronal behaviours. The fields of computational neuroscience and Systems Biology have developed as separate disciplines for a long time and only fairly recently has the neurosciences started to incorporate methods from Systems Biology. Metabolic Control Analysis (MCA), a Systems Biology tool, has not been used in the neurosciences. This study aims to further bring these two fields together, by testing the feasibility of an MCA approach to analyse the Hodgkin-Huxley model. In MCA it is not the parameters of the system that are perturbed, as in the more traditional sensitivity analysis, but the system processes, allowing the formulation of summation and connectivity theorems. In order to determine if MCA can be performed on the Hodgkin-Huxley model, we identified all the discernable model processes of the neuronal system. We performed MCA and quantified the control of the model processes on various high-level time invariant system observables, e.g. the action potential (AP) peak, firing threshold, propagation speed and firing frequency. From this analysis we identified patterns in process control, e.g. the processes that would cause an increase in sodium current, would also cause the AP threshold to lower (decrease its negative value) and the AP peak, propagation speed and firing frequency to increase. Using experimental inhibitor titrations from literature we calculated the control of the sodium channel on AP characteristics and compared it with control coefficients derived from our model simulation. Additionally, we performed MCA on the model&rsquos time-dependent state variables during an AP. This revealed an intricate linking of the system variables via the membrane potential. We developed a method to quantify the contribution of the individual feedback loops in the system. We could thus calculate the percentage contribution of the sodium, potassium and leak currents leading to the observed global change after a system perturbation. Lastly, we compared ion channel mutations to our model simulations and showed how MCA can be useful in identifying targets to counter the effect of these mutations. In this thesis we extended the framework of MCA to neuronal systems and have successfully applied the analysis framework to quantify the contribution of the system processes to the model behaviour.

AFRIKAANSE OPSOMMINMG: Die Hodgkin-Huxley-model, wat in 1952 ontwikkel is, was een van die eerste modelle in rekenaarmagtige neurowetenskap en is vandag steeds een van die bes-bestudeerde neuronmodelle. Hoewel daar vele modelle bestaan met &rsquon meer uitvoerige sisteembeskrywing as die Hodgkin-Huxley-model gee dié model nietemin &rsquon akkurate beskrywing van verskeie hoëvlak-sisteemverskynsels. Die twee velde van sisteembiologie en neurowetenskap het lank as onafhanklike dissiplines ontwikkel en slegs betreklik onlangs het die veld van neurowetenskap begin om metodes van sisteembiologie te benut. &rsquon Sisteembiologiemetode genaamd metaboliese kontrole-analise (MKA) is tot dusver nog nie in die neurowetenskap gebruik nie. Hierdie studie het gepoog om die twee velde nader aan mekaar te bring deurdat die toepasbaarheid van die MKA-raamwerk op die Hodgkin-Huxley-model getoets word. In MKA is dit nie die parameters van die sisteem wat geperturbeer word soos in die meer tradisionele sensitiwiteitsanalise nie, maar die sisteemprosesse. Dit laat die formulering van sommasie- en konnektiwiteitsteoremas toe. Om die toepasbaarheid van die MKA-raamwerk op die Hodgkin-Huxleymodel te toets, is al die onderskeibare modelprosesse van die neurale sisteem geïdentifiseer. Ons het MKA toegepas en die kontrole van die model-prosesse op verskeie hoëvlak, tydsonafhanklike waarneembare sisteemvlak-eienskappe, soos die aksiepotensiaal-kruin, aksiepotensiaal-drempel, voortplantingspoed en aksiepotensiaal-frekwensie, gekwantifiseer. Vanuit hierdie analise kon daar patrone in die proseskontrole geïdentifiseer word, naamlik dat die prosesse wat &rsquon toename in die natriumstroom veroorsaak, ook sal lei tot &rsquon afname in die aksiepotensiaal-drempel (die negatiewe waarde verminder) en tot &rsquon toename in die aksiepotensiaal-kruin, voortplantingspoed en aksiepotensiaalfrekwensie. Deur gebruik te maak van eksperimentele stremmer-titrasies vanuit die literatuur kon die kontrole van die natriumkanaal op die aksiepotensiaaleienskappe bereken en vergelyk word met die kontrole-koëffisiënte vanuit die modelsimulasie. Ons het ook MKA op die model se tydsafhanklike veranderlikes deur die verloop van die aksiepotensiaal uitgevoer. Die analise het getoon dat die sisteemveranderlikes ingewikkeld verbind is via die membraanpotensiaal. Ons het &rsquon metode ontwikkel om die bydrae van die individuele terugvoerlusse in die sisteem te kwantifiseer. Die persentasie-bydrae van die natrium-, kalium- en lekstrome wat tot die waarneembare globale verandering ná &rsquon sisteemperturbasie lei, kon dus bepaal word. Laastens het ons ioonkanaalmutasies met ons modelsimulasies vergelyk en getoon hoe MKA nuttig kan wees in die identifisering van teikens om die effek van hierdie mutasies teen te werk. In hierdie tesis het ons die raamwerk van MKA uitgebrei na neurale sisteme en die analise-raamwerk suksesvol toegepas om die bydrae van die sisteemprosesse tot die modelgedrag te kwantifiseer.


Thinking about the nerve impulse: A critical analysis of the electricity-centered conception of nerve excitability

Nerve impulse generation and propagation are often thought of as solely electrical events. The prevalence of this view is the result of long and intense study of nerve impulses in electrophysiology culminating in the introduction of the Hodgkin-Huxley model of the action potential in the 1950s. To this day, this model forms the physiological foundation for a broad area of neuroscientific research. However, the Hodgkin-Huxley model cannot account for non-electrical phenomena that accompany nerve impulse propagation, for which there is nevertheless ample evidence. This raises the question whether the Hodgkin-Huxley model is a complete model of the nerve impulse. Several alternative models have been proposed that do take into account non-electrical aspects of the nerve impulse and emphasize their importance in gaining a more complete understanding of the nature of the nerve impulse. In our opinion, these models deserve more attention in neuroscientific research, since, together with the Hodgkin-Huxley model, they will help in addressing and solving a number of questions in basic and applied neuroscience which thus far have remained outside our grasp. Here we provide a historico-scientific overview of the developments that have led to the current conception of the action potential as an electrical phenomenon, discuss some major objections against this conception, and suggest a number of scientific factors which have likely contributed to the enduring success of the Hodgkin-Huxley model and should be taken into consideration whilst contemplating the formulation of a more extensive and complete conception of the nerve impulse.

Keywords: Action potential Electromechanical pulse Hodgkin-Huxley model Nerve impulse Neuroscientific models Signal propagation.

Copyright © 2018 The Authors. Published by Elsevier Ltd.. All rights reserved.


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