1. Fractional Differential Equation for the Analysis of
Electrophysiological Recordings
Darío M. Domínguez1, Mariela Marín2
Mathematics Department, Universidad Militar Nueva Granada, Bogotá,
1
Colombia
Biophysical Laboratory, Centro Internacional de Física, Bogotá, Colombia
2
Email: fracumng@umng.edu.co
Abstract: The use of fractal analysis is a study for times series of
macrophage’s macroscopic ion currents signals. Hurst coefficients (H) and fractional
dimension were calculated on the time series in the zone between peak and steady
state currents of the pulse assuming white noise. We show that H, is different from
0.5, indicating that the time series cannot be considered white noise. H is not only
different but below 0.5 implying an anti persistent pattern. In addition, we show that
fluctuation of the ion currents IH versus voltage curves fit an equation where IH (V) =
f (V, α, m, d) for a voltage V, α associates with fractional calculus and m, d fit the
model to the voltage domains studied. Fitting by α fractional confirms that the
phenomenon has memory and we suggest that α values are associated with the
complexity of the current.
Keywords: Hurst, fractal, ion current, fractional, memory.
1. Introduction
Time series recordings by electrophysiological techniques allow the detection
of capacitive currents; currents passing through ion channels and cellular
membrane potentials and they are use for functional characterization of
electrical properties and ion channels of the cellular membrane. There are
two basic ways of recording information from the cellular membrane with
electrophysiological techniques. One in which only a small patch of the
membrane is recorded (single channel recordings) and in which the current
through one or few channels is observed (ion current). Another in which all
the cellular membrane is recorded (whole cell recordings) and the currents
through many ion channels are observed (macroscopic currents). In single
channel, the data set is analyzed to determine energy barriers, velocity of
transition between open and close states and ion channel conductance.
The use of fractal statistical analysis is mainly in single channel currents,
particularly on experimental data coming from excitatory plasma membranes
as those found in neurons. This type of analysis assumes that the ion current
recorded is a phenomenon that has memory, thus ion channel fluctuations can
be assumed as a series of large number of sub states, switching between sub
states may vary in time and may be linked and therefore previous states are
important Bassingthwaighte1.
Fractal analysis predicts that the apparent constant velocity during
channel opening and closing will vary inversely against the scale of the time
series observed, therefore short time series reveal fast velocities whereas long

2. Domínguez DM, Marín M.
time series reveal slow velocities Liebovitch2; Liebovitch and Sullivan3;
Mandelbrot4.
Much less, other works has made on models and with whole cell
recordings (total currents). In this work we have applied a non-linear times
series techniques from the data set provided by Biophysical Lab. (CIF
Bogotá). Time series signal of ion currents from macrophage-like cells, were
recorded using the whole cell configuration of the patch clamp technique
Hamill5. Our main interest is to try to classify these series and to analyze if
their dynamical behaviors are correlated in some way. First, a preliminary
study carried out with the aim of characterizing those times series in terms of
long-term memory (R/S analysis) and the fractal dimension calculus (D).
Here we adjust the model through the solution of a differential equation of
fractional order.
2. Time Series of the Electrophysiological recordings
Outward currents of Macrophages: Cell membranes solve the
problem of exchanging ions across them using integral membrane proteins.
Among these proteins, ion channels are the most efficient, diffusing ions in
favor of their electrochemical potential. Ion channels are ubiquitous in cells,
are complex proteins that span the cell membrane, gate their aqueous pore in
response to voltage differences, ligands or mechanical stimuli and are
involved in functions related with action potential propagation, muscle
contraction and cell signaling.
In this work, we studied the time series of the macrophage’s ion
current signals. Macroscopic outward currents Io through cellular membranes
from cells of the immune system, using the whole cell configuration of the
patch clamp technique by Hamill5. The time series are outward current
elicited in response to voltage steps from -40 mV to 90 mV; in 20 mV
increments (see Figure 1). Each data set contains 3800 points, which
corresponds to ionic flow at this potential. Time series are the study to use
fractal analysis Glöckle and Nonnenmacher6, Leibovitch7.
Figure 1. Ionic currents present in control J774.1 cells. Outward current elicited in
response to voltage steps from a holding potential of -60 mV. To values ranging from
-40mV to 90 mV, in 20 mV increments.

3. Fractional Differential Equation for the Analysis of Electrophysiological Recordings -
The data sets which we used was purchased by Biophyisical Lab (CIF). The
data sets are recording by the electrophysiologial thecnique in whole cell
configuration for detection of macroscopic currents. The Figure 1 represents
the data sets after transfomation. Each data set cell (whlose cell recording)
corresponds to 10 ion current siganls that contains 3800 points each one.
3. Data Analysis
3.1 R/S Analysis
Rescaled Rang analysis (R/S analysis) is the tool to study long-term memory
and fractality, first introduced by Hurst (1951) in hidrology and then
Mandelbrot (1983) said that R/S analysis is more powerful tool in detecting
long range dependence than the conventional analysis like correlation
analysis, variance ratios and spectral analysis. In this method, one measure of
culmulative deviations from the mean of the serie is changing with the time.
It has found that, for some time series, the dependence of the R/S on the
number of data point follows an empirical power law described as
(R/S)=(R/S)0nH , where(R/S)0 is a constant, n is the time index for periods of
different longth, and H is the Hurst exponent (R/S)n is defined as
max1≤ r ≤ n A ( t , n ) − min1≤ r ≤ n A ( t , n ) (1)
⎛R⎞
⎜ ⎟=
⎝ S ⎠n 1τ
∑ ( s (t ) − S n )
2
n t =1
Where A(t,n) is the cumulative departure of the time seires s(t) from the time
t +n
A (t, n ) = ∑ ( s (i ) − s )
average over the time interval n : s 2
n n
i =t
The Hurst exponent, 0 ≤ H ≤ 1, is equal to 0.5 for random, white noise series,
<0.5 for rough anticorrelated series, and >0.5 for positively correlated series.
In this work, the recordings obtained were converted to Microsoft
Excel 4.0 and then transferred to Benoit, 1.3, Fractal Analysis System
(TruSoft Int'l Inc., St. Petersburg, FL, USA) to estimate H. We consider three
controls cells for the analysis. Each cell had 10 times series for current
elicited in response to voltage steps from a holding potential of -60 mV. The
results for these cells, are summarised on table 1. As it can see, the time
series between I peak (Ipeak) was the peak current determined at 42 ms and
stacionay current (Iss), was the mean current at the end of the pulse that was
790-830 ms for outwards currents, normally assumed to be white noise. We
obtained values different and bellow of 0.5 for all times studied, suggesting
that the time series does not follow a random pattern but that the phenomena
has memory, wich indicates that the series are antipersistent. This result
indicates that the fluctuations of the time series studied correlated negatively.

4. Domínguez DM, Marín M.
Voltage (mV) Hcell(1) Hcell(2) Hcell(3)
-90 0.347 0.336 0.3
-70 0.221 0.169 0.224
-50 0.197 0.176 0.185
-30 0.233 0.2 0.187
-10 0.275 0.233 0.213
10 0.285 0.265 0.246
30 0.273 0.251 0.269
50 0.286 0.279 0.273
70 0.294 0.293 0.287
90 0.336 0.288 0.316
Table 1. Estmated H, using Benoit Software for three cells in the differents potentials.
3.2 Fractal Dimension
The IOut recordings (Figure 1), show fluctuations that appear to be self-
similar. Calculation of fractal dimension as well as of H by rescaled range
analysis were performed in the interval between times to Ipeak and Iss where
signals are usually assumed to be white noise (42−790 ms for IOut),
corresponding to a non-stationary process. Once the H were calculated, the
fractal dimension is defined as D=2-H, (the results not showed).
If the fractal dimension is D=1.5, or the Hurst coeficient is H=0.5, and the
correlation is 0 then the local flows are completely random.
The value of H is associated with the extent of the fluctuation in the flow in
this area, associated with the gating of the channels and then IH is defined as
the value of coefficient Hurst at intervals to their respective voltage.
Once the IH values are calculated, we plot them against Vm (see Figure 2).
The curves obtained have the same pattern as that found when the ratio Iss to
the apparent Ipeak was plotted versus voltage (not showed).
0,5
0,5
0,4
0,4
0,3
0,3
IH
IH
0,2
0,2
0,1
0,1
0,0
0,0
-100 -80 -60 -40 -20 0 20 40 60 80 100
-100 -80 -60 -40 -20 0 20 40 60 80 100
V (mV)
0,5
V (mV)
0,4
0,3
IH
0,2
0,1
0,0
-100 -80 -60 -40 -20 0 20 40 60 80 100
V (mV)
Figure 2. Fluctuation currents IH vs. voltage (mV). Outward currents for three
control cells.

5. Fractional Differential Equation for the Analysis of Electrophysiological Recordings -
4. Fractional Differential equation: IH as a function of voltage
Definition: The Hurst coefficient is like the fluctuation current in the time
series. So, IH(V) as H is a value different from 0.5 indicates a process with
memory (Figure 3). Therefore, the curves obtained were fitted with a model
that also has memory, in other words a model derived from fractional
calculus Glockle and Nonnenmacher6; Carpentieri and Mainardi8;
Bassingthwaighte et al.1; Picozzi and West9; Vargas et al.10; Vargas et al.11;
Metzler and Klafter12. The model with the best fit gives the following
equation:
⎛ π ⎞⎞
⎛
⎛ π ⎞⎞
⎛
⎜ ( mV + d ) cos ⎜ ⎟ ⎟
2
I H (V ) = cos ⎜ ( mV + d ) sin ⎜ ⎟ ⎟ + 0.31
⎝α ⎠⎠
⎝
(2)
e
α ⎝α ⎠⎠
⎝
This fractional equation (2) fits the IH as a function of voltage for IO
(Figure 3). The curve which fit is significant by chi2 analysis (p-value <0.01).
The parameters m and d in this equation are scaling parameters. V is voltage,
and α is a parameter that related with diffusion. For IO, at voltages between -
90 to 90 mV, the found values were: m was 0.024 d was 4.5 and α was 1.36.
The IH vs voltage curve is a function of the theoretical solution of the
differential fractional equation
Dα I H (V ) = I H (V ) (3)
13
When H<0.5, one has α = 2 / ( 2 H 0 + 1) Darses and Saussereau .
The solution of the fractional differential equation is
I H (V ) = − J α I H (V ) (4)
For the Laplace transform
Sα
L ( Ι H (V ) ) = (5)
1+ Sα
For the Laplace inverse transform
)
(
Sα Sα
1 1 1 iπ − iπ
∫ e 1 + S α dS = 2π i Ha e 1 + S α dS + α te α + te α
∫
I= SV SV
2π i Br
= dα ( t ) + gα ( t ) thus
1 ⎧ t cos⎜ α ⎟ ⎛ it sin ⎜ α ⎟ − it sin ⎜ α ⎟ ⎞ ⎫
⎛π ⎞ ⎛π ⎞ ⎛π ⎞
⎪ ⎪
gα ( t ) = ⎨e + e ⎝ ⎠ ⎟⎬ (6)
⎜e
⎝⎠ ⎝⎠
⎜ ⎟
α⎪ ⎠⎪
⎝
⎩ ⎭
We obtain
⎛π ⎞
⎛ π ⎞⎞
⎛
2 (7)
t cos ⎜ ⎟
gα ( t ) = ⎝α ⎠
cos ⎜ t sin ⎜ ⎟ ⎟
e
α ⎝α ⎠⎠
⎝

6. Domínguez DM, Marín M.
The other function dα(t) is not considered, because there are not data
from the experiments.
0,5 0,5
0,4 0,4
0,3 0,3
IH
IH
0,2 0,2
0,1
0,1
0,0
0,0
-100 -80 -60 -40 -20 0 20 40 60 80 100
-100 -80 -60 -40 -20 0 20 40 60 80 100
V (mV)
V (mV)
0,5
0,4
0,3
IH
0,2
0,1
0,0
-100 -80 -60 -40 -20 0 20 40 60 80 100
V (mV)
Figure 3. Experimental data fit by gα
5.Conclusions
In this paper we first examined a process of gating memory because the
Rescaled Range Analysis (R/S analysis) shows that there is an antipersistent
through the calculation of the Hurst coeficient, H<0.5 in all time series,
indicating that is a memory process. In addition, the fractal analysis shows
that on any scale of the process the gating is not random. The fractional
analysis suggests that the underlying process in the macroscopic scale of
gating is a complex process and does not follow the laws of simply random.
Then there is a determinism in gating of ionic currents when the process
begins. Finally, determining the behavior of the model, made possible to
study how the macrophages acts when it is infected, which will be our future
study.
Acknowledgment
This work was supported by Grant: 1123240520182 from the Programa
Ciencias Básicas, COLCIENCIAS. Universidad Militar Nueva Granada and
Centro Internacional de Física, Bogotá, Colombia.

7. Fractional Differential Equation for the Analysis of Electrophysiological Recordings -
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