PII: S0169-2607(00)00076-6

Computer Methods and Programs in Biomedicine 63 (2000) 47–54

Simulation of axonal excitability using a Spreadsheet

template created in Microsoft Excel

Angus M. Brown *

Department of Neurology, Uni6ersity of Washington School of Medicine, Box

356465

, Seattle, WA

98195

6465

, USA

Received 23 August 1999; received in revised form 23 January 2000; accepted 25 January 2000

Abstract

The objective of this present study was to implement an established simulation protocol (A.M. Brown, A

methodology for simulating biological systems using Microsoft Excel, Comp. Methods Prog. Biomed. 58 (1999)

181–90) to model axonal excitability. The simulation protocol involves the use of in-cell formulas directly typed into

a spreadsheet and does not require any programming skills or use of the macro language. Once the initial spreadsheet

template has been set up the simulations described in this paper can be executed with a few simple keystrokes. The

model axon contained voltage-gated ion channels that were modeled using Hodgkin Huxley style kinetics. The basic

properties of axonal excitability modeled were: (1) threshold of action potential ﬁring, demonstrating that not only

are the stimulus amplitude and duration critical in the generation of an action potential, but also the resting

membrane potential; (2) refractoriness, the phenomenon of reduced excitability immediately following an action

potential. The difference between the absolute refractory period, when no amount of stimulus will elicit an action

potential, and relative refractory period, when an action potential may be generated by applying increased stimulus,

was demonstrated with regard to the underlying state of the Na

and K

channels; (3) temporal summation, a

process by which two sub-threshold stimuli can unite to elicit an action potential was shown to be due to conductance

changes outlasting the ﬁrst stimulus and summing with the second stimulus-induced conductance changes to drive the

membrane potential past threshold; (4) anode break excitation, where membrane hyperpolarization was shown to

produce an action potential by removing Na

channel inactivation that is present at resting membrane potential. The

simulations described in this paper provide insights into mechanisms of axonal excitation that can be carried out by

Keywords

Axon; Hodgkin Huxley; Ion channel; Microsoft Excel; Modeling; Simulation; Spreadsheet

www.elsevier.com/locate/cmpb

1. Introduction

This article is an extension of a previous study

that described in detail how to execute simula-

tions of biological systems using a spreadsheet

template created in Microsoft Excel [1]. The focus

* Tel.: +1-206-6168278; fax: + 1-206-6858100.

E-mail address

[email protected] (A.M. Brown)

PII: S0169-2607(00)00076-6

A.M. Brown

Computer Methods and Programs in Biomedicine

63 (2000) 47 –54

of this present study was to implement that simu-

lation protocol to describe some fundamental

properties of axonal excitability based on the

squid giant axon containing a fast Na

current,

which results in the upstroke of the action poten-

tial, a delayed rectiﬁer K

current, which results

in membrane repolarization, and an ohmic leak

current that determines resting membrane poten-

tial. The simulation uses the rate constants

derived by Hodgkin and Huxley [2] to describe

the voltage dependence of ion channel behavior,

although they have been updated to reﬂect mod-

ern conventions [1,3]. This simulation serves two

purposes. Firstly, it allows the user to conduct

simulations to investigate mechanisms of axonal

excitation in the squid giant axons. However,

other preparations can be modeled simply by

changing the appropriate rate constants and con-

ductances. Secondly, the user can study and

graphically display the underlying properties of

ion channels, such as activation, inactivation and

the resulting conductance changes, to see how

those properties determine axonal behavior.

Familiarity with the previous paper [1] is essen-

tial, as this present study uses it as a stepping

stone to demonstrate ‘real life’ scenarios. This

paper will be appreciated most by those who are

interested in carrying out interactive simulations

of ion channel behavior, but who do not wish to

expend the time and money necessary to learn

programming. The simulation involves using in-

cell formulas in which rows and columns of new

data are generated from key parameter values

typed into the spreadsheet, and solving a set of

equations based on those parameters. The fea-

tures of Excel that make it ideal for this purpose

are a user friendly interface, ﬂexible data han-

dling, in-built mathematical functions and instan-

taneous charting of data. The objective of this

study was to use an established, easy to use

simulation protocol to demonstrate key features

of axonal excitability determined by ion channel

properties.

2. Computational method

Full details of the computational method illus-

trated in this present study have appeared previ-

ously [1]. The objective of the simulation was to

determine how the membrane potential of a

model squid giant axon containing I

and I

responded to a variety of stimuli, i.e. it is a

current clamp simulation where changes in mem-

brane potential were modeled in response to con-

stant current injection. Each stimulus paradigm

was designed to illustrate an individual property

of axonal excitability. Brieﬂy, the simulation is

carried out by (1) setting the initial membrane

potential, (2) setting the amplitude and duration

of current injection, (3) sequentially solving a

series of equations describing the rate constants,

(in)activation parameters, conductances, currents

and ﬁnally, change in membrane potential, respec-

tively, based on the initial values input by the user

in the ﬁrst two steps. In current clamp simulations

the change in membrane potential (V) over time is

described by:

total

(1)

In the simulation described in this paper

total

leak

inj

(2)

where

=120m

h(− V+ 50) mAcm

−2

(3)

=36n

(− V− 77) mAcm

−2

(4)

leak

=0.3( −V −59.4) mAcm

−2

(5)

and I

inj

is the injected current input by the user.

The (in)activation parameters are described by

m(h, n)=

a+ b

(6)

rate constants for am, bm, an, bn, ah and bh can

be found elsewhere [1,3].

The spreadsheet template is illustrated in Fig. 1.

This template is identical to the one described

previously ([1], see Fig. 4 p. 186) and all the

expressions used in the calculations are the same

([1], see Table 2 p. 184). Each column contains the

solution of a separate voltage dependent parame-

A.M. Brown

Computer Methods and Programs in Biomedicine

63 (2000) 47 –54

ter (column B contains am, column C contains

bm, etc.). Column A contains the time parameter

where the time increment in 0.04 ms. The user

inputs the amplitude of injected current in cell B2.

Column P contains the data referring to injected

current. Thus, the duration of the current injec-

tion can be altered by increasing or decreasing the

number of rows in which it appears. In the ﬁrst

row of calculations (row 4 on the spreadsheet) the

parameters are solved based on the initial value of

V input by the user in cell B1. In this instance it

is set to −70 mV. The calculations are carried

out sequentially from cell B4 to cell Q4. The I

total

(see Eq. (2)) is then used to calculate the new

membrane potential (V) in cell R4. The voltage

dependent parameters in the next row are solved

starting at cell B5 sequentially from B5 to Q5

using the new value of V in cell R4, and so on,

sequentially down the rows. In the following sim-

ulations only the value of current injection, initial

membrane potential or duration of current injec-

tion, were altered once the template has been set

up. These parameters can be altered as necessary

by changing values in cell B1, B2 or in column P,

respectively, to conduct the simulations described

below. The data of interest can be studied graphi-

cally, by plotting the time parameter (column A)

against the column containing the appropriate

data (e.g. for g

, the data is in column K).

. Action potential threshold

The threshold, or critical depolarization, for

action potential ﬁring is considered to be the

membrane potential above which regenerative de-

polarization occurs, resulting in the ﬁring of an

action potential. Fig. 2A illustrates the calculated

time course of a uniformly propagated action

potential and the underlying Na

and K

con-

ductance changes. Note that the current injection

of 10 mA (C) causes an increase in g

resulting in

depolarization of the membrane. This is followed

by a delayed increase in g

, which results in

repolarization of the membrane towards rest. The

increase in g

outlasts the duration of the stimu-

lus, a factor that is important in temporal summa-

Fig. 1. Spreadsheet template used to calculate V. The value for V in cell B2 is the initial membrane potential and is used in the

calculations in row 4. The value of V in cell R4 is then used as the voltage parameter in the calculations in row 5 to calculate the

new value of V. Calculations are carried out sequentially down the rows where the value for V calculated in column R of the

previous row is used in the calculations of the subsequent row to calculate the new value of V, and so on. Rows 14–503 have been

omitted to save space. At row 504 it can be seen that the value in column P has changed from 0 (no injected current) to 10 mA

reﬂecting the value of I

inj

in Cell B2.

A.M. Brown

Computer Methods and Programs in Biomedicine

63 (2000) 47 –54

Fig. 2. Stimulus evoked action potential. (A) The action

potential (bold) results from an increase in g

, which depolar-

izes the membrane, followed by a delayed increase in g

which results in repolarization of the membrane. An after-hy-

perpolarization occurs due to g

outlasting g

. (B) The total

conductance is plotted illustrating that when g

is larger than

there is an increase in membrane conductance (arrow)

favoring a depolarization of the membrane. The faint line

illustrates sub-threshold stimulation for comparative purposes.

evoke the action potential.

ization increases. Finally the current injection is

sufﬁciently large (2.5 mA) to cause the all or

nothing action potential, due to the larger Na

conductance outbalancing the K

conductance.

Chronaxie is deﬁned as the minimum duration of

twice the amplitude of rheobase needed to pro-

duce an action potential and is illustrated in Fig.

3B. This illustrates that both duration and ampli-

tude of current injection are important in deter-

mining if threshold is reached.

Threshold, however, is not a ﬁxed value but

varies depending on axonal membrane potential

at the time of current injection (20 mA), a feature

illustrated in Fig. 4. It can be seen that for the

same amplitude of injected current, a larger g

occurs at more hyperpolarized potentials ( −80

mV) than at depolarized membrane potentials

(− 60 mV; Fig. 4B). This is because hyperpolariz-

ing the axonal membrane removes Na

channel

inactivation, resulting in more available Na

channels. The current pulse recruits these addi-

Fig. 3. Rheobase and chronaxie. (A) The threshold current

required to elicit an action potential is 2.5 mA (the largest

illustrated current). Increasing sub-threshold pulses result in

increasing membrane potential changes. (B) Chronaxie is the

minimum stimulus duration required to elicit an action poten-

tial when applying a current of twice the rheobase (5 mA). In

this case the duration was 6.2 ms.

tion (see later). Fig. 2B illustrates the conductance

changes critical for action potential ﬁring. In

terms of conductance, threshold is deﬁned as the

potential at which inward Na

conductance is

greater than the outward K

conductance leading

to regenerative membrane depolarization. If the

current is larger than Na

current the mem-

brane potential will return to rest (faint trace),

however if the Na

current is marginally larger

that the K

current (arrow) membrane potential

will become unstable and produce an action po-

tential. This is explained in the Hodgkin cycle of

excitability where depolarization of the

membrane leads to increased Na

permeability

resulting in net inﬂux of Na

, which leads to

further membrane depolarization and so on [4].

Rheobase is deﬁned as the minimum amount of

current required to produce an action potential [3]

and is illustrated in Fig. 3A. As the amplitude of

current injection increases so membrane depolar-

A.M. Brown

Computer Methods and Programs in Biomedicine

63 (2000) 47 –54

Fig. 4. The initial resting membrane potential determines the

proﬁle of the action potential. (A) A hyperpolarized mem-

brane potential (−80 mV, bold trace) results in a larger action

potential than a more depolarized membrane potential (−60

mV, faint trace). (B) This is due to removal of Na

channel

inactivation resulting in larger g

due to more available Na

channels, and decreased g

(not shown). (C) Current injection

of 20 mA for 5 ms.

Immediately after the ﬁrst action potential h is

close to 0 indicating the majority of Na

channels

are inactivated, and n is close to 1, indicting that

the majority of K

are open, a fact reﬂected in

the hyperpolarized membrane potential immedi-

ately after the action potential. Thus there are two

factors which result in decreased excitability im-

mediately following an action potential; (1) the

majority of Na

channels are in the inactivated

state and unavailable for opening; and (2) the

Fig. 5. Refractory period. (A) The absolute refractory period is

the period immediately following an action potential when no

amount of current injection will elicit a second action poten-

tial. Top panel: the ﬁrst action potential was elicited by a

current of 40 mA. Increasing this stimulus amplitude by an

order of magnitude failed to elicit a second action potential.

Middle panel: this is due to increased g

reﬂected in the value

of n near to 1, and the Na

channels still being in the

inactivated state, reﬂected by the value of h close to 0. Bottom

panel: the current injection proﬁle. (B) The relative refractory

period occurs immediately after the absolute refractory period

when a second stimulus can elicit a second action potential but

the stimulus intensity must be increased. Top panel: a second

action potential can be elicited if an increased current is

injected (200 mA). The ﬁrst action potential was elicited by 40

mA. The faint trace shows that a second current pulse of 40 mA

failed to elicit an action potential. Middle panel: The values of

n and h are returning towards rest when the second pulse is

injected. Bottom panel: the current injection proﬁle.

tional Na

channels which are available at −80

mV but are inactivated at −60 mV, and hence

the increased g

. Refractory period

Refractoriness is the period of decreased ex-

citability of an axon immediately after an action

potential. There are two types of refractoriness,

absolute and relative. The absolute refractory pe-

riod refers to the period immediately after an

action potential, when it is impossible to elicit a

further action potential no matter how much cur-

rent is injected. Fig. 5A illustrates this phe-

nomenon and its underlying mechanism. An

initial current injection of 40 mA for 1 ms elicits

an action potential. Repeating this stimulus 3 ms

later fails to elicit a second action potential, even

when current injection is increased by an order of

magnitude to 400 mA. The reason for this lack of

excitability is shown in the middle panel which

displays the K

channel activation parameter n,

and the Na

channel inactivation parameter h.

A.M. Brown

Computer Methods and Programs in Biomedicine

63 (2000) 47 –54

Fig. 6. Temporal summation and anode break excitation. (A)

The faint trace shows the effect of only one stimulus, which

fails to elicit an action potential. An action potential is elicited

when two sub-threshold stimuli are injected (bold trace). (B)

Injection of an anodal current pulse (bottom panel) results in

membrane hyperpolarization (top panel). ‘Breaking’ the an-

odal pulse results in membrane excitation culminating in an

action potential. Middle panel: This is due to removal of Na

channel inactivation (arrow indicates where h increases to-

wards 1 indicating reduction in Na

channel inactivation),

and decrease in g

reﬂected in a decrease in n towards 0.

. Temporal summation

This term usually refers to synaptic transmis-

sion where sub-threshold postsynaptic potentials

overlap and accumulate to drive the membrane

potential past threshold resulting in a postsynap-

tic action potential. Here the basic principal is the

same but the inputs are direct current injection

into the axon. This effect is demonstrated in Fig.

6A and shows how two sub-threshold stimuli of

40 mA amplitude and 200 ms duration delivered 1

ms apart can elicit an action potential (bold

trace), whereas a single stimulus of the same

dimensions fails to elicit an action potential (faint

trace). The stimulus-induced changes in mem-

brane conductance outlast the duration of the

stimulus, and it is possible for the conductance

changes brought about by the ﬁrst stimulus to

sum with the conductance changes elicited by the

second stimulus if the interval between the two

stimuli is sufﬁciently small. The summed conduc-

tances can then drive the membrane potential past

threshold.

. Anode break excitation

This phrase refers back to the terminology used

in the 1940s where anodal current injection results

in a hyperpolarizing of the membrane potential.

The anode break refers to removal of a hyperpo-

larizing stimulus which gives rise to an action

potential. Thus, an action potential can ﬁre even

after a current injection that results in membrane

hyperpolarization. The middle panel illustrates

the events underlying this phenomenon. As the

anodal current is injected, h, the Na

channel

inactivation parameter increases towards 1 signal-

ing that Na

channel inactivation is being re-

moved and more Na

channels are available for

opening. Conversely the activation parameter n,

for K

channels decreases towards 0 resulting in

decreased K

conductance. During the period of

anodal polarization there is a reduced outward

current and an increased Na

current. Thus

release of the polarization results in a membrane

depolarization that rapidly becomes regenerative

and results in threshold being reached and an

action potential ﬁring.

majority of K

channels are open resulting in

hyperpolarization of the membrane.

The relative refractory period occurs after the

absolute refractory period and refers to the fact

that a second action potential can be elicited but

a larger current must be injected. This is illus-

trated in Fig. 5B. Delaying the interval between

the two stimuli to 8 ms permits current injection,

albeit of an increased amplitude (200 mA), to elicit

a second action potential (top trace). This is be-

cause the increased interval allows h to return

towards resting levels resulting in more Na

channels available for opening, and n decreases

(middle trace) resulting in less K

channels being

open resulting in a decreased level of membrane

hyperpolarization. A larger current pulse must be

injected to overcome the fact that neither n nor h

has returned to their resting levels.

A.M. Brown

Computer Methods and Programs in Biomedicine

63 (2000) 47 –54

. Repetiti6e ﬁring

In 1928 Adrian outlined a theory of repetitive

ﬁring [5], to describe how stimuli of increasing

strength resulted in increased frequency of action

potential ﬁring. The objective of the Hodgkin

Huxley model was to describe membrane perme-

ability changes associated with a single action

potential, and predicts repetitive ﬁring over a

limited range of frequencies in the squid giant

axon. Adding hypothetical K

conductances and

altering the voltage dependent properties of K

currents extends the range of ﬁring frequencies to

more accurately describe the repetitive ﬁring pat-

terns seen in squid giant axon [6,7]. However, for

demonstration purposes it is viable to use the

original Hodgkin and Huxley model to simulate

repetitive ﬁring. Fig. 7 illustrates the property of

increased stimulus amplitude resulting in in-

creased action potential frequency. (A) Injecting a

current stimulus of 8 mA for 120 ms elicited a

pattern of seven evenly spaced action potentials

with inter-spike intervals of 17.8 ms. (B) Increas-

ing the stimulus amplitude to 20 mA decreased the

inter-spike interval to 11.8 ms, increasing the

number of action potentials to 11.

3. Discussion

In this paper current clamp simulations of ex-

citability of a model of the squid giant axon, a

classic electrophysiological preparation, are de-

scribed. The large diameter of the giant axon of

the squid Loligo allowed experimenters to insert

microelectrodes inside the axon to record ‘intra-

cellular’ responses of the axon and thus made it

an ideal preparation in which to study excitation

[8]. Subsequent investigation using the voltage

clamp technique allowed experimenters to control

the voltage of a piece of membrane and determine

ion movements at ﬁxed voltages [9 –12]. These

data culminated in the classic description of Na

and K

permeability changes during excitation

and conduction in squid giant axon [2]. Hodgkin

and Huxley derived a series of equations that

accurately described the changes in permeability

in Na

and K

responsible for an action poten-

tial. Their model also predicted some basic prop-

erties of an excitable membrane, such as

refractoriness and anode break excitation de-

scribed in this paper. Why go to the trouble of

repeating the Hodgkin and Huxley model? The

model is still valid almost 50 years after its initial

description, and is widely used to describe behav-

ior of voltage-gated ion channels [13]. A great

advantage of the model is that it can be success-

fully applied to other preparations such as the

R15 neuron of Aplysia, which displays a more

complicated pattern of ﬁring that the squid axon

[14]. The advantage of the protocol described in

this paper is that it does not require any program-

ming knowledge, which in today’s Windows based

environment, is costly and very time consuming.

It does not require the expense of purchasing

specialized software such as Neurosim [15], Neu-

ron [16] or Genesis [17]. It uses a spreadsheet on

a standard desktop PC which almost all biologists

know how to use, even if at an elementary level.

Such is the advance in technology that computa-

tion of an action potential which originally took 8

Fig. 7. Repetitive ﬁring. (A) Injection of a 120 ms current

pulse of 8 mA (faint trace in C) produces repetitive ﬁring in the

model axon. (B) Increasing the current amplitude to 20 mA

(bold trace in C) increases the number of action potentials

from 7 to 11. The scale bar represents 40 mV in A and B.

A.M. Brown

Computer Methods and Programs in Biomedicine

63 (2000) 47 –54

h on a hand held calculator can now be done in

under1sona400MHzPentium II computer with

64 MB RAM using the protocol described here.

References

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