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Journal of Neuroscience Methods 122 (2002) 97 /108
www.elsevier.com/locate/jneumeth
Measurement of diffusion parameters using a sinusoidal
iontophoretic source in rat cortex
Kevin C. Chen, Charles Nicholson Department of Physiology and Neuroscience, New York University School of Medicine, 550 First Avenue, New York, NY 10016, USA
Received 10 June 2002; received in revised form 16 September 2002; accepted 16 September 2002
Abstract
A new method was developed to extract diffusion parameters in brain tissue using a sinusoidal iontophoretic point source of
tetramethylammonium operated at different frequencies. The resulting steady state oscillating extracellular concentration of this
probe molecule was continuously monitored using an ion-selective microelectrode located about 100 mm from the source. Because
the probe molecules must diffuse through the extracellular space (ECS), the oscillating concentration at the recording location will
develop a phase lag and an amplitude attenuation relative to the sinusoidal source. These two components of the signal can be
analyzed to determine the tortuosity factor l and the ECS volume fraction a . The method also measures the nonspecific clearance
rate constant k . In brain slices this reflects washout of diffusing molecules. Values of a (0.189/0.05) and l (1.679/0.08) obtained
from this frequency method in rat cortical slices were similar to those obtained by the real-time iontophoretic method employing a
square pulse source. The relative merits of the frequency method compared to the pulse method are discussed.
# 2002 Elsevier Science B.V. All rights reserved.
Keywords: Extracellular space; Tortuosity; Volume fraction; Clearance; Ion-selective microelectrode
1. Introduction
Many brain functions and processes involve diffusion
of neurotransmitters, neuromodulators, and metabolic
substrates through the brain extracellular space (ECS;
Agnati et al., 2000; Barbour and Ha¨usser, 1997; Egelman and Montague, 1999; Nicholson, 2001; Nicholson
and Rice, 1991; Rusakov and Kullmann, 1998b; Sykova´,
1997; Zoli et al., 1999).
When the length scale of interest in the brain ECS
extends over several cellular dimensions, extracellular
diffusion is often described macroscopically (Nicholson
and Phillips, 1981; Nicholson et al., 2000; Nicholson,
Abbreviations: ECS, extracellular space; ISM, ion-selective
microelectrode; TMA, tetramethylammonium; ACSF, artificial
cerebrospinal fluid; RTI, real-time iontophoresis; SD, spreading
depression.
Corresponding author. Address: Department of Neuroscience and
Physiology, New York University, New York, NY 10016, USA. Tel.:
/1-212-263-5421; fax: /1-212-689-9060
E-mail address: [email protected] (C. Nicholson).
2001), in the sense that the classic diffusion equation still
applies in the brain after certain modifications. The
modifications can be encapsulated in two locally averaged parameters: the ECS volume fraction a , and the
tortuosity l . This macroscopic approach is relevant also
to discussion of extrasynaptic diffusion of neurotransmitters (Barbour and Ha¨usser, 1997; Barbour, 2001;
Rusakov and Kullmann, 1998a,b; Rusakov, 2001).
The ECS volume fraction a is defined as the ratio of
the extracellular volume available for the diffusion of a
molecule that remains within this compartment to the
total tissue volume, and l is defined as l/(D /D )1/2,
where D is the effective (apparent) diffusion coefficient
of a given molecule in the brain and D the diffusion
coefficient of the same molecule in a free solution. The
tortuosity l is a measure of how the structure of the
brain ECS hinders extracellular diffusion. Conceptually,
l is often interpreted as the relative increase in diffusion
path length compared to that in free solution (Nicholson
and Sykova´, 1998; Nicholson et al., 2000; Nicholson,
2001; Rusakov and Kullmann, 1998a). But it should be
noted that, from the definition of l, it also may
0165-0270/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 0 2 7 0 ( 0 2 ) 0 0 2 9 9 - 6
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K.C. Chen, C. Nicholson / Journal of Neuroscience Methods 122 (2002) 97 /108
incorporate other effects such as additional interstitial
viscosity arising from extracellular matrix molecules
(Rusakov and Kullmann, 1998a).
A constant k (also designated as k ?) is often included
in the diffusion equation to account for various
clearance processes (e.g., cellular uptake, loss across
blood vessels or into cerebrospinal fluid or washout
from brain slices) that eliminate diffusing molecules
from the ECS. The three diffusion parameters of the
ECS (a , l, and k ) have been studied in vivo as well as in
vitro by the real-time iontophoretic (RTI) method (e.g.
Nicholson and Phillips, 1981; Nicholson, 1992, 1993;
Rice and Nicholson, 1991; Nicholson and Sykova´, 1998;
Vorˇ´ısˇek and Sykova´, 1997). In this method, the extracellular probe cation, tetramethylammonium (TMA ;
molecular weight 74 Da), is released by iontophoresis
from a micropipette by a pre-defined current pulse, of
about 50 s duration, and the locally averaged ECS
concentration of TMA at a known distance from the
source is monitored by an ion-selective microelectrode
(ISM). Reviews of results obtained with the RTI method
have been given by Nicholson et al. (2000), Nicholson
(2001), Nicholson and Sykova´ (1998), Sykova´ (1997)
and Sykova´ et al. (2000).
In this paper, we modify the RTI method by
sinusoidally modulating the strength of the iontophoretic current with time using a waveform characterized by
a fixed magnitude and frequency. The physical picture
behind this intensity-modulated diffusion approach is
conceptually similar to the multiple scattering of timeresolved near-infrared light in tissues, or the so-called
photon-migration technique (Sevick-Muraca et al.,
1997), widely used in optical biodiagnosis (Fishkin et
al., 1991, 1995). For simplicity, the previous RTI
method employing a square pulse diffusion source will
be referred to as the ‘pulse method’, and the modified
protocol described here will be referred to as the
‘frequency method’.
order elimination mechanism that accounts for all
nonspecific clearance processes. Nonlinear uptake mechanisms (such as Michaelis-Menten kinetics) are not
considered here, because such processes preclude an
analytical solution to the diffusion equation (Nicholson,
1995). Convective interstitial bulk flow (Rosenberg et
al., 1980) is absent in the regions of interest.
2.1. Constant source
Currently, the most widely used diffusion paradigm is
the release of a substance from a point source into the
ECS at a constant rate Q (mol s 1) beginning at time
zero and ending at time tp. The analytical diffusion
behavior has been well characterized (Nicholson, 1992,
1993, 2001; Nicholson and Phillips, 1981). Assuming
spherical symmetry, the concentration of the diffusing
molecules in the ECS at a radial distance r away from
the point source localized at r/0 for 00/t B/tp is
described as
C(r; t)
We start by describing the basic equations used to
extract the diffusion parameters. First, we state our
assumptions. The brain tissue is considered as an
infinitely large porous medium with isotropic, homogeneous, macroscopic properties. The validity of this
assumption in slices will be discussed later. For the
typical recording distance between source and ISM (/
100 mm), the iontophoresis micropipette can be regarded
as a point source, consequently, spherical geometry
applies. The macroscopic diffusion properties of the
medium are embodied in the three constant parameters,
a , l , and k , as described previously. The probe molecule
(usually TMA ) is assumed to remain predominately in
the ECS. Any loss of molecules occurs through a first-
8pD+ r
pﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
r
er k=D erfc pﬃﬃﬃﬃﬃﬃﬃﬃ kt
2 D+ t
pﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃ
r
r k=D+
erfc pﬃﬃﬃﬃﬃﬃﬃﬃ kt ;
e
2 D+ t
(1)
in which erfc() is the complementary error function. To
obtain the effect of a pulse of finite duration, a timedelayed version of Eq. (1) is subtracted from the
original. This is the theoretical basis for the many
experimental studies using the RTI pulse method.
Assuming the diffusing probe molecules are ions with
a charge number z, the source term Q can be characterized by
Q
2. Theory
Q
nt I
;
zF
(2)
where I (amp) is the iontophoretic current, F is the
Faraday constant (96485 C mol 1), and nt is the
transport number (fraction of the current I that carries
the probe ion).
Because we will be mainly interested in the steadystate behavior, it is more appropriate to rewrite Eq. (1)
in a form that separates the steady and transient
components,
pﬃﬃﬃﬃﬃﬃﬃ+ﬃ
Q
Q
C(r; t)
er k=D +
3=2
4paD r
2p aD+ r
pﬃﬃﬃﬃﬃ
kr2
+
u2
r=2 D t
g
0
4D+ u2
e
du:
(3)
K.C. Chen, C. Nicholson / Journal of Neuroscience Methods 122 (2002) 97 /108
In the right-hand side of Eq. (3), the first term is the
steady-state component and the second term (the
integral term) is the transient component of the solution,
with u being the dummy integration variable. At longer
times, the transient term decreases to zero because of the
diminishing upper limit of the integral r/2(D t ). Thus
Eq. (3) is useful for situations approaching a steady state
(i.e., larger t ). Looking at the first component, it is seen
that, in steady state, the radial concentration distribution monotonically decays according to the product 1/r
and exp(/r(k /D )). The inverse decay with 1/r is
caused by the expanding diffusion space as r increases;
clearance processes can further expedite the radial decay
through the exponential term exp(/r (k /D )).
2.2. Sinusoidal source
A new approach is considered here where the source
magnitude is periodic in time and is modulated with a
sine function Q sin(vt/o ), where v is the angular
frequency and o the initial phase offset. The analytical
solution C (r , t) for such a sinusoidal source without the
clearance term (k /0) has been given by Carslaw and
Jaeger (1959, Section 10.4, Eq. 12). We derived the
corresponding solution for a finite k as
sﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ﬃ Q
g
+
r g =D
C(r; t)
e
sin vtr
o
4paD+ r
D+
Q
g
e
0
sin v t
2
r
4Du2
o du;
2.3. Sinusoidal source with net positive output
To obtain a reliable iontophoretic release of TMA from the micropipette, the iontophoretic control unit
can be programmed to output an iontophoretic current
defined by
I(t)I1 I2 sin(vto);
(7)
where I1, I2 are constants and I1 E/I2 /0, thus ensuring
a net positive current at all times. Similarly, for an
anionic probe, a negative applied current pattern would
be used. Combining the results for the constant and
sinusoidal diffusion sources, the steady-state expression
for C (r, t) corresponding to the iontophoretic current
described by Eq. (7) is
pﬃﬃﬃﬃﬃﬃﬃ+ﬃ
nt I1
C(r; t)
er k=D
4pzF aD+ r
qﬃﬃﬃﬃ
g
r
nt I2
D
e
sin(vtuo)
4pzF aD+ r
S Asin(vtuo);
(8)
with the symbols S and A representing the two
components of the steady-state concentration level; the
first associated with the constant term identical to the
steady-state component of the pulse method and a
second associated with the amplitude of the oscillating
component.
2.4. Nikolsky equation
pﬃﬃﬃﬃﬃ
kr2
r=2 D+ t u24D+ u2
2p3=2 aD+ r
99
(4)
To relate the acquired voltage signals to the ionic
concentration, the ISM must be calibrated through use
of an empirical modification of the Nernst equation,
called the Nikolsky equation (Ammann, 1986; Nicholson, 1993),
in which g and g are defined as
V V0 s[log10 (C ki )log10 (C0 ki )];
1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
g [ v2 k2 9k]:
2
where V is the voltage (after subtraction of the reference
barrel voltage, see below) corresponding to C , s is the
slope of the ISM, ki is the interference from other cation
species, and V0 and C0 are a pair of reference values
obtained by calibrating the ISM with a standard
concentration during the experiment.
9
(5)
Here the integral term in Eq. (4) is again the transient
component of the solution. For a sufficiently large time,
the limit of the upper integral also approaches zero and
the transient term vanishes. Then the concentration C (r ,
t) at r oscillates with the same angular frequency v as
the diffusion source at the origin, but the phase angle
lags behind the source by
sﬃﬃﬃﬃﬃﬃ
g
u r
;
(6)
D+
which is hereafter referred to as the phase lag. If k /0
(as in the case of diffusion in dilute agar gel), g /g /
v /2 and Eq. (6) is simplified to u/r(v /2D ).
(9)
3. Materials and methods
Adult female Sprague/Dawley rats (/250 g) were
anesthetized with sodium pentobarbital (55 mg kg1
i.p.) and then decapitated, in accordance with local
IACUC regulations. After removal of the brain from the
skull, 400 mm-thick paracoronal sections were cut with a
vibrating microtome (VT1000S, Leica; Nußloch, Germany) in an ice-cold artificial cerebrospinal fluid
100
K.C. Chen, C. Nicholson / Journal of Neuroscience Methods 122 (2002) 97 /108
(ACSF), followed by a recovery period of 1 /2 h in
oxygenated (95% O2 /5% CO2) ACSF at local room
temperature, 27 8C. The composition of the ACSF was
(in mmol per liter): 115 NaCl, 5 KCl, 35 NaHCO3, 1.25
NaH2PO4, 1.3 MgCl2, 1.5 CaCl2, 10 D-glucose, and 0.5
TMA chloride. The TMA was added to the ACSF to
provide a calibration reference for the ISM measurements in the experiments and to elevate the baseline
TMA in the tissue above the ambient K interference
level (Rice and Nicholson, 1991). The ACSF had an
osmolarity of /303 mosmol kg1 and a pH of /7.4.
Following recovery, slices were transferred to an interface recording chamber (Model BSC-BU with Hass top,
Harvard Bioscience, Holliston, MA) and the lower
surface of the slice continuously perfused at 2 ml min 1
with the ACSF while the upper surface was bathed in
humidified 95% O2 /5% CO2 gas. The slice experiments
were also conducted at room temperature (/27 8C).
ISMs sensitive to TMA were made from doublebarreled theta glass (Warner Instrument, CT, USA) as
described elsewhere (Nicholson, 1993). One barrel was
silanized before filling the tip with Corning 477317 ionexchange resin (currently available as IE 190 from WPI,
Sarasota, FL). The ion-sensing barrel was then backfilled with 150 mM TMA chloride while the reference
barrel was filled with 150 mM NaCl. Both barrels were
connected via Ag/AgCl wires to buffer and subtraction
amplifiers (Model IX2-700 with N /0.001 and N /0.1
headstages, Dagan Corporation, Minneapolis, MN). All
signals were recorded against a remote indifferent Ag/
AgCl/KCl electrode that was connected to the ground of
the recording system. Local DC potentials were recorded by the reference barrel and continuously subtracted from those recorded on the ion-sensing barrel.
Amplified ion and reference voltage signals were lowpass filtered to remove frequencies above 6 Hz using a 4pole Bessel filter (CyberAmp 380, Axon Instruments,
Union City, CA) before being sent to an A/D converter
(PCI-MIO-16E-4, National Instruments, Austin, TX).
The digitized data were transferred to a personal
computer (PC) where they were analyzed with a LabVIEW (version 5.1, National Instruments) program
WANDA (Chen, unpublished). ISMs were calibrated
before and after each experiment in a series of TMA solutions ranging from 0.5 to 8 mM in doubling
concentrations against a constant ionic background (3
mM KCl and 150 mM NaCl) using the fixed interference method (Nicholson, 1993). Calibration data
were fitted to the Nikolsky equation to determine the
electrode slope and interference. Typical TMA -sensitive ISMs had a slope of 59 mV per decade and an
interference (ki) of 0.02 mM within the concentration
range tested. The TMA diffusion source consisted of a
micropipette prepared from theta glass. The shank was
slightly bent before back-filling with 1 M TMA . A
microelectrode array was made by gluing together an
iontophoretic pipette and an ISM with a tip separation
80 /100 mm. The iontophoretic micropipette was connected to an iontophoresis unit (ION-100, Dagan)
controlled by the PC and the WANDA program
through a D/A channel on the A/D board referred to
above. A schematic experimental setup and ISM configuration is shown in Fig. 1(a).
Sinusoidal diffusion experiments were first performed
in 0.3% agar (Agar Noble, Difco, Detroit, MI) gel, made
up in 150 mM NaCl, 3 mM KCl, and 0.5 mM TMA ,
to obtain the TMA free diffusion coefficient D and the
iontophoretic transport number nt. The array of electrodes was then lowered vertically into the center of the
slice (nominally 200 mm beneath the superficial slice
surface) to repeat the same sinusoidal diffusion experiments. To convert the voltage signals into ionic concentrations, a reference voltage corresponding to 0.5
mM TMA was obtained by submerging the ISM in the
bathing ACSF. A constant bias current of /20 nA was
applied continuously to maintain a steady transport
number. At the onset of measurements, a sinusoidal
current waveform defined by Eq. (7) and generated with
the WANDA program and A/D converter was applied
to the iontophoretic electrode through the iontophoresis
unit. Values of the applied iontophoretic currents (I1
and I2) remained unchanged throughout the same series
of experiments. Typical applied currents were I1 /I2 /
100 nA with an initial phase offset o of /0.5p. A
sequence of sinusoidal diffusion experiments was performed with the following frequencies (Hz), f/0.01,
0.02, 0.04, 0.05, 0.08, 0.1, 0.2, and 0.4. Then v /2pf.
For each v , both ionic and DC potential signals were
recorded simultaneously at 10 samples per second from
t /0 until signals of constant amplitude and phase were
achieved in the recorded signals for more than 30 min.
The phase lag u and the oscillation amplitude A of
the recorded signals in response to a specific angular
frequency v were determined by fitting the time series of
the ISM concentration measurements in the steady state
with C (t)/S/A sin(vt/u/o ) using the Levenberg /
Marquardt algorithm (More´, 1978), which is known to
be a robust nonlinear curve-fitting algorithm. The
measured u was then plotted as u2 versus v . The
theoretical relationship is
u2 r2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
( v2 k2 k):
2D+
(10)
When the medium was agar (a /1, l /1, k /0), Eq.
(10) simplified to u2 /(r2/2D )v and the value of D in
the agar could be directly obtained from the slope of the
fitted straight line for a given r , which was obtained by
visual measurement using a calibrated compound microscope. Upon knowing D , the transport number nt
could be calculated from the amplitude A . In the brain
slice, because the v used was much greater than k , we
K.C. Chen, C. Nicholson / Journal of Neuroscience Methods 122 (2002) 97 /108
101
Fig. 1. (a) Experimental setup for the ISM array and diffusion experiments. The TMA -selective double barrelled electrode was glued to a slightly
bent iontophoretic microelectrode. Figure shows calibrating solution over agar or brain; during brain measurements this is replaced with humidified
O2/CO2. Ind. is indifferent (ground) electrode. (b) A typical diffusion curve measured by ISM when a TMA -pulse experiment was performed in the
rat brain (cerebral cortex, layer IV) with a pulse duration of 60 s. The array spacing was r/83 mm and the applied currents are I1 /100 nA and I2 /
0. (c) A representative oscillating TMA potential waveform in the same brain region. The time axis was transformed into the phase angle, vt/u .
The acquired signal was obtained with the iontophoretic microelectrode emitting a sinusoidal waveform defined by I1 /I2 /100 nA, o //0.5p, and
f /0.02 Hz. The local DC potential from the reference barrel of the ISM (gray curve) was superimposed to define the phase lag u .
fitted the simplified Eq. (10) as u2 /(r2/2D )(v/k ).
The value of D was determined from the slope of the
fitted straight line and k was obtained from the vertical
intercept extrapolated to v /0. The tortuosity l was
calculated as (D /D ). The value a was subsequently
obtained from the curve fitting of the amplitude A at
fixed r versus v upon substituting in the transport
number nt previously determined in agar.
A total of 10 rats were used with two or three slices
from each rat. For each slice, the pulse method was first
performed several times and the acquired data were
analyzed using the program WALTER developed by
Nicholson (unpublished). The procedure for the pulse
method has been well documented in the papers of
Nicholson and Sykova´ et al., and will not be repeated
here. After the pulse experiments, the complete series of
sinusoidal diffusion measurement was performed with
an ascending set of frequencies. When switching to a
different frequency, a sufficient time ( /30 min) was
allowed for the TMA baseline concentration in the
slice to be restored. At the conclusion of the sinusoidal
diffusion experiments, the pulse method was run two or
three times again to confirm that the tissue ECS
properties had not changed.
4. Results
Diffusion experiments using a sinusoidal waveform
were conducted in cortical layers III /V, but there was
no statistically significant variation in the fitted parameters with location. The parameters are summarized in
Table 1. The average values were a /0.189/0.05
(mean9/SEM, n/21), l /1.679/0.08 (n/25), and
k /0.0259/0.005 s 1 (n /25). In the pulse experiments,
we obtained a /0.209/0.03, l /1.659/0.04, and k /
0.0109/0.002 for n /11. A mean value of the free
D (/10.84 /10 6 cm2 s1), measured in agar at
27 8C, was used to calculate l. Parameters such as a
and l obtained from both the pulse and frequency
methods were similar to those in previous studies
in rat cortex, either in vivo or in vitro (Hrabe˘tova´ and
Nicholson, 2000; Hrabe˘tova´ et al., 2002; KumeKick et al., 2002; Lehmenku¨hler et al., 1993; Mazel et
al., 1998; Pe´rez-Pinzo´n et al., 1995). But the k
obtained by the frequency method appeared to be
larger than previously reported values in similar 400
mm slices using the pulse protocol (see Table 1 for
comparisons). Possible reasons will be explored in the
Section 5.
102
K.C. Chen, C. Nicholson / Journal of Neuroscience Methods 122 (2002) 97 /108
Table 1
Fitted diffusion parameters (mean9SEM) for TMA in rat cortical slices from this work in comparison with previously reported values in similar
slices
This work
This work
Hrabe˘tova´ et al. (2002)
Hrabe˘tova´ and Nicholson (2000)
Pe´rez-Pinzo´n et al. (1995)
Kume-Kick et al. (2002)
a
l
k (10 2 s 1)
Remark
0.1890.05 (21)
0.2090.03 (11)
0.2290.03
0.2590.01
0.1890.01
0.2490.01 (42)
1.6790.05 (25)
1.6590.04 (11)
1.6590.02
1.6690.05
1.6290.04
1.6990.01 (42)
2.4690.42 (25)
1.0490.21 (11)
0.7390.22
1.2090.30
0.5390.13
1.9090.10 (42)
a
b
b
b
b
b
Numbers in the parenthesis indicate the total number of measurements. a, sinusoidal waveform source; b, pulse source.
Fig. 1(c) shows a typical time series (expressed in units
of phase angle, vt/o ) from the sinusoidal diffusion
measurements in cortical layer IV at a distance r /80
mm away from the source. Because the DC potential,
recorded from the ISM reference barrel, synchronized
with the intensity-modulated iontophoretic source, it
was superimposed with the oscillating ionic signal to
show the phase difference between the two waves. It can
be seen that the recorded ionic oscillation, after a
transient rise from the onset, established a regular
oscillation at the same frequency as the DC wave but
with a finite phase lag. The transient rise lasted about 2
min and reflected the temporal behavior of the integral
term in Eq. (4), which vanished to zero within the time
scale of r2/D /0.5 min. Fig. 1(c) only demonstrates
regularly oscillating behavior with constant amplitude
and non-zero phase lag; it does not necessarily show that
the acquired concentration wave is also sinusoidal. To
examine this, a segment of the ISM potential waveform
was converted to units of concentration using Eq. (9)
and fitted with C (t )/S/A sin(vt/u/o ). The fitting
Fig. 2. Fitting the oscillating diffusion response at steady state from
Fig. 1 by C (t ) /S/A sin(vt/u/o ). With f /0.02 Hz and o //908,
the best fit (solid line) according to the Levenberg /Marquardt
nonlinear fitting algorithm corresponded to S /1.771 mM, A /
0.267 mM, and u/138.558 ( :/0.77p). When the data C (t ) were
displayed as [C (t )/S ]/A versus sin(vt/o ), the result was an ellipse
with its two main axes orientated along the diagonals of the figure.
Denoting the lengths of the two principle axes by a and b , the phase
lag u is related to the axis lengths by a/(1/cosu ) and b /(1/
cosu ).
result is displayed in Fig. 2 by plotting the normalized
[C (t)/S ]/A , i.e., sin(vt/u/o ), versus sin(vt/o ) to
form a Lissijou figure. The normalized display forms an
ellipse with the principle axes in the diagonal directions.
The lengths of the two axes then determine the phase lag
u . Note that if u /0, p, 2p, . . ., the ellipse will collapse
to a straight line in either diagonal direction.
The sensitivity and accuracy of the fitting was
assessed in Fig. 3 by 100 fits of randomly chosen ISM
waveform segments from Fig. 1(c). Quantile plots of the
three fitted parameters (u , A , and S ) showed the
characteristic shapes associated with normal distributions (Chambers et al., 1983). In accord with this, the
means of the three parameters [u /(138.719/0.84)8, A /
(0.2659/0.004) mM, and S /(1.7729/0.008) mM] were
very close to their corresponding median values in the
quantile plots (138.68, 0.267 mM, and 1.772 mM,
respectively). Although it appears from Fig. 3(a) that
the fitted u, A , and S are scattered randomly, the
datasets for all three parameters, in fact, have very
narrow distributions, as indicated by the very small
standard deviations, indicating that u , A , and S can be
determined with high accuracy without being seriously
distorted by transients from background noise. Fig. 2
and Fig. 3 confirmed that the acquired diffusion wave
not only possessed a constant phase lag, but also was
sinusoidal, as expected.
By repeating the same waveform acquisitions at the
same location with various frequencies, one can construct two plots like those shown in Fig. 4 to reveal how
the phase lag u and the amplitude A change with the
angular frequency v. It appears that the experimental
data agree well with the theory. From the plot in Fig.
4(a), the diffusion parameters D and k were determined by the slope and intercept of the fitted line.
Subsequently, a was determined from Fig. 4(b). Note
that the maximum distinguishable phase lag from
experiments is 2p. Any value for u , when expressed in
the form of 2m p/f (in which m /0, 1, 2, 3,. . .), can
only be identified as the remainder f. This situation
happens when v exceeds a threshold. For k /1/102
s 1, D /4/10 6 cm2 s 1, and r/100 mm, u will
become larger than 2p only when f /0.5 Hz. Therefore,
K.C. Chen, C. Nicholson / Journal of Neuroscience Methods 122 (2002) 97 /108
103
the range of f used in our experiments (0.01/0.4 Hz)
guarantees that the fitted u falls within [0, 2p].
During some experiments, spontaneous spreading
depression (SD) was encountered. SD is characterized
by a slowly propagating wave of cellular depolarization
that causes a transient suppression of all neuronal
activity (Lea˜o et al., 1947; Martins-Ferreira et al.,
Fig. 4. Representative examples of the data analyses for diffusion
parameters in the cortex. (a) Plot of u2 versus f (/v /2p). With r/101
mm, D and k were determined from the slope and intercept of the
fitted linear line, u2 /(r2/2D )(v/k ). See text for more explanation.
(b) Plot of the oscillation amplitude A versus f (/v /2p). After D and
k were obtained for a given r , the term exp (/r (g /D )) could be
calculated. Fitting the oscillation amplitude A (according to Eq. (8))
versus v then determined the best fit of the ratio nt/a . In agar (a/1),
this ratio corresponded to nt; in brain, upon substituting the known nt
determined from agar experiments, this ratio yielded a .
2000). Our experience was that SD usually happened
when the same slice was recorded repetitively and/or
when a high frequency was applied.
Fig. 5 shows a record of extracellular TMA concentration ([TMA ]e) and DC extracellular potential during an SD. Concomitant changes in the
[TMA]e and DC extracellular potential clearly indi-
Fig. 3
Fig. 3. Statistical analysis of the fitting data (u , A , and S ). A total of
100 fittings with the randomly chosen ISM waveform segments taken
from Fig. 1 was performed, and the fitted data were displayed in three
dimensions in (a). The fitted parameters (expressed as means9/SEM)
were u/(138.719/0.84)8, A/(0.2659/0.004) mM, and S/(1.7729/
0.008) mM. (b) Quantile plots of u , A , and S . Each fitted parameter
was sorted in ascending order and plotted against its rank as a fraction
of the total data points. The quantile of a dataset is the fraction (or
percentage) of the data points less than or equal to the given value
(Chambers et al., 1983). The median, corresponding to 0.5 (or 50%)
quantile, is a measure of the center of a distribution. The medians for
u , A , and S at 50% quantiles are 138.68, 0.267 mM, and 1.772 mM,
respectively.
104
K.C. Chen, C. Nicholson / Journal of Neuroscience Methods 122 (2002) 97 /108
Fig. 5. Chart records of ISM responses and extracellular potential recorded in cortical slices exhibiting a spontaneous SD. Source frequency f /0.04
Hz and array spacing r/102 mm. The ISM ionic signals have been converted into TMA concentration. The smooth gray lines indicate the assumed
average behaviors of the waveforms. Except for the two points A and G, the transient u during SD (points B /F) was approximately estimated by
fitting just one complete cycle within the enclosed dashed-line boxes. The phase lags were (A) 104.18, (B) 109.28, (C) 117.68, (D) 124.58, (E) 119.48, (F)
114.28, and (G) 104.08. Inset in upper figure shows the possible changing paths of l versus a during SD.
cated the occurrence of SD. The grey lines are not
numerically averaged from the oscillating wave, but
were simply inserted to help indicate the assumed
smooth behaviors. The shape of the negative shift in
the DC potential could be recognized to be characteristic of SD; it had an inverted saddle shape, with a small
rising ‘notch’ separating two negative maxima, of which
the second maximum was often more negative and
prolonged than the first one (Herreras and Somjen,
1993). The onset of the negative shift in the DC potential
occurred at about the same time as when the [TMA]e
started to rise, but recovered well before the [TMA]e
fell back to baseline. The recovery of DC potential
occurred at the same time that the elevated [TMA]e
reached a maximum at approximately twofold the
baseline concentration, indicating that the local ECS
volume fraction during SD in our experiment shrank to
about half of its original size. The SD in Fig. 5 induced a
DC potential shift about /12 mV and lasted about 2.5
min, exhibiting a larger amplitude and longer duration
than has been observed in some other neocortical slices
(Vila´gi et al., 2001), but was very similar both in shape
and amplitude to the SD episodes shown in Tao (2000)
and Tao et al. (2002) in the same cortical region. In
addition to the shift of the mean DC potential during
SD, a larger oscillation amplitude of the DC potential
was also observed, which probably reflects the increased
tissue resistivity as cellular swelling reduces the volume
of the ECS.
During SD, there is a massive inward movement of
NaCl across cell membranes that has to be accompanied
by water (Kraig and Nicholson, 1978; Nicholson et al.,
1981), causing cells to swell and so reducing the ECS
volume. This change in a is expected to be accompanied
by an increase in tortuosity (Chen and Nicholson, 2000;
Kume-Kick et al., 2002; Nicholson and Rice, 1991) and,
consequently, the phase lag u . Even though our analysis
was based on steady state conditions, we made a rough
estimate of the u during SD in Fig. 5. The u before SD
was found to be 104.18 (at point A), extended to a
maximum 124.58 (at point D) during SD, and fully
recovered back to the same pre-SD value (104.08 at
point G) after SD. For f /0.04 Hz and r/102 mm,
assuming that k was significantly smaller than
v during SD, the tortuosity l was estimated to increase
from 1.65 at the steady state before SD to a maximum
1.98 during SD, while a decreased from 0.2 to 0.1. The
extent of the SD-induced ECS shrinkage reported here is
similar to that observed by Tao et al. (2002) in the same
cortical region, although a more severe ECS shrinkage
(a decreased from 0.23 to 0.05) was reported by
Ande˘rova´ et al. (2001), together with a greater increase
in l (from 1.67 to 2.29) during SD. The studies reported
here and by Tao et al. were in slices while Ande˘rova´ et
al. used an intact animal preparation for their SD
studies and this likely accounts for the differences in
results.
K.C. Chen, C. Nicholson / Journal of Neuroscience Methods 122 (2002) 97 /108
5. Discussion
5.1. Model assumptions
We first examine the validity of some of the underlying assumptions. The first concern is the regional
heterogeneity and anisotropy in the brain tissue.
Although the rat cortex is usually considered isotropic
(Nicholson and Sykova´, 1998; Nicholson, 2001), heterogeneity and anisotropy occur in other brain regions
(Rice et al., 1993; Mazel et al., 1998; Sykova´ et al.,
2002). As TMA molecules diffuse through different
brain regions, the local a , l and k may vary, a
possibility that was not taken into account in our
model. Another concern is whether an unperturbed
boundary condition can be assumed at r / 8/ in view
of the long diffusion times involved in the frequency
method. This question is especially relevant in the slice
model.
Because we used an interface recording chamber, the
slice upper surface was in contact with a humidified gas
stream, which constituted a zero flux condition, while
the diffusing TMA reaching the lower slice surface
would be swept away by the ACSF flowing underneath,
constituting a zero concentration condition. The thickness of the slice was 400 mm. The diffusion source was
positioned approximately at the center of the slice so the
distance to either boundary was 200 mm. If the upper
boundary was perfectly reflecting and the lower was
perfectly absorbing, an image argument would indicate
that their respective contributions would cancel at the
center of the slice, whereas moving the electrodes
towards the lower boundary would lead to an apparent
increase in clearance. Model calculations were made
with a semi-infinite medium and an absorbing boundary
to simulate washout; these showed that the measured
value of k /0.025 s 1 can be accounted for by washout
but that the source and measuring electrodes would
have to be 77 mm from the lower surface. While some
error in placement is possible, an error of this magnitude
is unlikely. Consequently, it seems that other factors are
involved in the removal of TMA from the slice in the
interface chamber and this is presently being studied.
Nonetheless, so long as k is accurately determined, the
biophysical origin is not important for the extraction
and interpretation of a and l.
105
r (g /D )). As is evident in Eq. (8) and Fig. 4, the
conclusion to be drawn is that the higher the frequency,
the less the penetration of the oscillating wave; i.e., highfrequency source oscillations are rapidly damped out in
comparison with the low-frequency oscillations.
In addition to v and r , it is interesting to observe the
functional dependence of the phase lag u and the
amplitude A on all three diffusion parameters (a , D and k ). According to Eq. (8), u is related to the effective
diffusion coefficient D and the clearance constant k
through the lumped factor r (g /D ), whereas A
exponentially decreases with r (g /D ). The definitions of g and g are provided in Eq. (5). Furthermore, the amplitude A is inversely scaled by the ECS
volume fraction a , as expected. Considering the fact that
the released probe molecules must diffuse through the
ECS before reaching the recording site, it is clear that u
should be proportional to r/D . But it is more difficult
to recognize that u is also affected by the concentrationdependent clearance process (although it is reasonable
to expect that k can reduce A ). Because v and k have
the same physical units (s1), we suggest that the
sinusoidally fluctuating source can be viewed as a
source/sink term that alternates signs periodically.
Thus v characterizes the frequency of the alternation
for the source/sink term, whereas k characterizes the
rate of TMA elimination from the ECS by the
nonspecific linear removal process. The resulting interaction of this alternate source/sink term with the linear
clearance term is a compromised phase shift as described
by Eq. (10). The form (v2/k2)/k probably reflects
the fact that the alternating source/sink is only in phase
with the uptake term every half cycle.
Fig. 6 indicates that for the same v , the overall effect
of a nonzero k is to reduce u slightly. At a lowfrequency, the influence of k on diminishing u is more
5.2. Functional dependence of u and A
Eq. (8) provides an explicit relationship showing how
the local concentration C (r, t ) is related to the source
angular frequency v . This equation indicates that in the
steady state the [TMA]e at r should oscillate sinusoidally with the same frequency as the source, but lag
behind with a phase angle u . Moreover, the amplitude
of the oscillation is diminished by the factor exp(/
Fig. 6. Effect of k on the theoretical curve for u2 versus f (/v /2p).
For a k much smaller than v , Eq. (10) is simplified to u2 :/
r2(2D )(v/k ) and the curve approximates a straight line. For a
given r , the D and k can be respectively determined from the slope of
the straight line and the intercept extrapolated to v /0. Comparing to
the curve of k/0, the effect of a finite k is to reduce u2 by r2(2D )(k )
at the same v . When k increases to become comparable to or larger
than the v , nonlinear fitting procedure is required to fit D and k .
106
K.C. Chen, C. Nicholson / Journal of Neuroscience Methods 122 (2002) 97 /108
pronounced. This means that for an extremely lowfrequency wave source or a pronounced k , such that
v /k , the phase lag will be more suppressed by k .
5.3. Determination of D and k
If the source frequency is much larger than the value
of k in the medium, the theoretical plot of the squared u
versus v will be linear. This is the case seen in our data,
as shown in Fig. 4(a), even though the experiments were
performed in slices. Because in Fig. 4(a) the data point
for the smallest v was still located in the linear region
(i.e., the smallest v used was still larger than k ), we
found that the nonlinear Levenberg /Marquardt curvefitting method failed to yield a convergent result for k .
Consequently, we elected to fit the u2 /v curve as a
straight line, and used the resulting intercept extrapolated from the fitted line to determine k . This procedure
is robust and efficient in the present study, but would
not be applicable if the k were sufficiently large that a
considerable nonlinear segment occurred in the u2 /v
curve. In this case, either a nonlinear fitting scheme
should be used to fit the data with Eq. (10), or one
should raise the source frequency further in order to
continue to fit the data linearly.
It is interesting to note that the frequency method
actually does not require knowledge of the interprobe
distance r to determine l and k . This is because fitting
u2 with v using Eq. (10) yields r2/(2D ) in agar, and r2/
(2D ) and k in brain. Since r is assumed unchanged
when the same ISM array is used in both agar and brain,
the tortuosity l can be directly obtained by the ratio of
the two slopes of the u2 /v curves in agar and brain,
respectively, without knowing r . The frequency method
will need the interprobe distance r only when we wish to
obtain D (or D ) alone or to calculate a . Because the
purpose of this work is to validate the frequency
method, we wish to obtain all three diffusion parameters
for comparisons with the pulse method. Because the
frequency method is unable to determine r and D
independently in agar, we used the same values of r
used in the pulse method, confirmed by visual measurement using a calibrated compound microscope.
5.4. Choice of frequency
In theory, it is desirable to employ as high a frequency
as possible to suppress the influence of k . A highfrequency source also shortens the time to reach steady
state (data not shown), making it easier for the
frequency method to capture the diffusion properties
during pathological events that occur on a time scale of
minutes (such as SD). However, a trade-off for increasing v is the rapidly-damped oscillation amplitude A .
Consequently, one must consider whether the ISM has
sufficient sensitivity to identify the true signals from the
noise fluctuations in the background, as well as adequate speed of response. Our experience is that the
oscillating ISM signals became indistinguishable from
background noise for f /0.5 Hz at a distance /100 mm
away from the source. However, higher frequencies can
be used if the distance is further shortened. Another
caution is that when a high v is used, the sampling
frequency must be increased accordingly to avoid signal
aliasing. To achieve this, the signal must be sampled at
least twice as rapidly as the frequency of the source. We
used a sampling rate of 10 point s 1, which is much
greater than twice the highest source frequency (0.4 Hz)
used in this work.
5.5. Potential application in studying SD
SD is a pathological event that only lasts 2/5 min.
Although the dynamic evolution of a can be monitored
by a tracer that remains predominantly in the ECS, such
as of TMA , measuring dynamic behavior of l during
SD is problematic. To date there is no reliable method of
studying how l changes during such a short-lived event
as SD. A recent attempt to measure l during SD used
the TMA pulse method (Ande˘rova´ et al., 2001) to
make one measurement. The l measured this way was
therefore a time-averaged value and did not provide
dynamic information. The frequency method proposed
here may have some potential in studying changes in l
corresponding to a during short-lived pathological
events. By employing a higher frequency, one can
make multiple measurements of the phase lag during
SD, and thereby obtain a rough but useful estimate of
the dynamic l. Most importantly, to extract D , the
pulse method uses the shape of the diffusion curve but
the frequency method employs the phase lag of the
recorded wave signals. Thus, the frequency method may
be less prone to the influence of an unsteady tracer
baseline concentration during SD. This is true as long as
the frequency of variations in the tracer baseline
concentration is much lower than the modulated source
frequency v .
Disparate l /a behavior during cell swelling/shrinkage via osmolarity challenge has been reported (KumeKick et al., 2002). One possible explanation is the
asymmetric modification in the cell shape during cell
swelling and shrinkage (Chen and Nicholson, 2000). The
inset in Fig. 5 shows the possible changing paths of l
versus a following ECS shrinkage and subsequent
restoration during SD. The difference between the two
paths is small. Considering the intrinsic errors in
determining u from a nonsteady state, it is possible
that the different paths depicted in Fig. 5 inset are an
artifact and that the l /a paths from A 0/D and D0/G
coincide with each other. However, if the changing paths
for l and a were indeed different, plots like the inset in
Fig. 5 would provide indirect evidence suggesting that
K.C. Chen, C. Nicholson / Journal of Neuroscience Methods 122 (2002) 97 /108
the ECS structure and connectivity does not respond to
the sudden modifications of the cell volume. That is to
say, l will change only when the change in a persists for
sufficient time to elicit a permanent modification in the
ECS structure. Thus, the frequency method may be
particularly suited to this kind of problem.
5.6. Merits and weakness of the frequency method
It is useful to compare this modified RTI method
employing a sinusoidal source (the frequency method)
with the previous paradigm employing a stepped source
(the pulse method). A distinct difference in the two
methods is in their very different time scales.
The frequency method needs to be performed under
steady-state conditions. Not only does the frequency
method take longer, but more measurements must also
be done at different frequencies. In contrast, the pulse
approach takes measurements on a much shorter time
scale.
However, the inconvenience of the frequency method
is compensated by the gains from other perspectives.
For instance, if the l and k are the only tissue
parameters of interest, the frequency method does not
require knowledge of the electrode array spacing r , as
shown above. In fact, to obtain l and k , the frequency
method does not require that the ISM has been
previously calibrated in various standard solutions,
nor must a reference potential be obtained with respect
to a known concentration in the tissue. Eq. (10) states
that to obtain l and k , the only experimental information needed is the phase lag u . But the phase lag can be
directly obtained from the recorded voltage wave as seen
in Fig. 1(c). This is because the ISM voltage potential
changes monotonically with its corresponding ionic
concentration (Eq. (9)). The maximum (minimum) of
the potential also corresponds to the maximum (minimum) of the ionic concentration. Hence, the phase lag
can be obtained by comparing the phase positions of the
peaks (or troughs) of the waveforms. Therefore, the
frequency method eliminates the requirement to convert
the acquired voltage signals into ionic concentrations by
Eq. (9). The pulse method can also yield both l and k
but the ISM slope must be determined (Nicholson and
Phillips, 1981). In practice, the slope of ISMs is almost
constant from one electrode to another so this is not a
major restriction. In both the frequency and pulse
methods, a full ISM calibration is required when a is
to be obtained.
The frequency method requires multiple measurements of u at different frequencies. At least two
different frequencies must be used. If the range of k
can be estimated in advance, the two different frequencies can be chosen high enough to ensure that u2
increases linearly with v (Fig. 6). Then the two sets of
measurements can determine r2/(2D ) and k . On the
107
other hand, if the experimental u2 /v curve appears to
be linear, one can conclude that the k must be smaller
than the smallest v used, and therefore get an upperlimit estimate of the clearance rate constant. If the u2 /v
curve appears to be nonlinear, one can still get an
estimate of k from the range of v where the curve
nonlincarity is located. Because of the multiple measurements and the fact that the slope is used, the frequency
method is also self-calibrating. Any inherent phase
errors in determing u that are associated with instruments or background noise will be cancelled upon
subtraction. In summary, the frequency method must
take measurements under steady-state conditions, and
must repeat the same measurements at more than one
frequency. Consequently, the frequency method can be
time-consuming. However, multiple measurements can
provide more information (how the phase lag and the
oscillation amplitude change with v and the estimate of
k from the u2 /v curve) and this may give the frequency
method some advantages.
A major potential advantage of the frequency method
is that it could be carried out with a phase-sensitive lockin amplifier (Meade, 1983), thus permitting the use of
extremely small source signals or alternatively, the
performance of measurements in certain types of very
noisy background. Thus the frequency method might be
advantageous in the intact animal where blood flow and
respiration increase noise substantially.
Acknowledgements
This research was supported by NIH grant NS 28642
from NINDS.
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