Memfractor: a common mathematical
framework for electronic circuit
elements with memory
René Lozi
Laboratory J.A. Dieudonné, University of Nice-Sophia Antipolis, France
Common work with
Mohammed-Salah Abdelouahab
University of Mila, Algeria
and
Leon O. Chua
Université de Berkeley, USA and Imperial College London, UK
The Memristor : the missing fourth element
43 years ago in his genuine paper “Memristor-the missing circuit
element," IEEE Transactions on Circuit Theory,18, 507-519,
[1971] , L.O. Chua predicted from a
theoretical point of view, the existence
of a missing passive circuit element
in generic electrical circuits componed
of resistor, capacitor and inductor.
He called this element memristor.
Such a physical device would not be
reported until 2008 when a physical
Image of 17 nano-memristors
model of a two-terminal hp device at
from HP laboratories
nano-scale, behaving has a memristor
was announced.
1 May 2008
"The Machine"
"The Machine"
Back to the Ohm’s law(s)
The voltage V standing between the poles of a battery in an electric
circuit with a resistor
R,
is linked to the intensity I of the current going
through the circuit by the famous Ohm’s law which any student learns
during physics lectures in any high school:
V = RI
Die galvanische Kette, mathematisch bearbeitet (1827)
When the voltage varies with respect to the time this relationship reads
v(t )  Ri (t )
L
If there is an inductor
instead of a resistor the relationship between
voltage (also called potential) and intensity reads:
d
v (t )  L i (t )
dt
Example
of my
physics’
handbook
in 1967
in
« Terminale
Math élem »
Magnetic
Flux
Physical origin
interaction between
magnetic field and the
shape of the electrical
circuit
Mathematical definition
integral of the electrical
potential (voltage) with
respect to the time
t
(t )   v() d 
t0
or
d (t )
v(t ) 
dt
Ohm’s law(s)
By integrating with respect to the time the Ohm’s law:
d
v (t )  L i (t )
dt
We get
(t )  Li (t )
We consider now the third classical passive element of electrical circuit:
the Capacitor.
CAPACITOR
The relationship
between voltage and
intensity for a
capacitor with
as capacity is
C
1 t
v(t )   i ()d 
C t0
which reads also,
considering the charge
q(t) :
1
v(t )  q (t )
C
The Memristor : the missing fourth element
In 1971, L.O. Chua, building
this chart flow, showed that
Resistor, Capacitor and
Inductor give a relationship
on each of the three sides
1
of the square.
v(t )  q (t )
C
Henceforth a relationship
was missing on the upper
side of the square, linking
flux and charge, i.e.
d (t )
RM (t ) 
dq(t )
(t )  Li (t )
v(t )  Ri (t )
voltage, Volt V
v
Resistor dissipates
Thermal Energy current, Ampere
A
i
R(v,i)=0
Capacitor
Stores
Electric Energy
L
C
L(,
i)=0
C(q,v)=0
R
Inductor
stores
Magnetic Energy
M
M ( , q)  0
q
charge, Coulomb C
Memristor
Stores
Information
φ
flux, Weber Wb
Axiomatic Definition of the Memristor
R(v, i)  0
M ( , q)  0
v
d dM (q) dq dM (q)
dq


i and i 
dt
dq dt
dq
dt
i
v  R ( q ) i

q  i
 v  R( ) i
v
 Memristor
  f ( , i) ̶
+
Memristor is defined by
a
State-Dependent
Ohm’s law
Experimental Definition of the
Memristor
i
+
v
̶
If it’s Pinched
It’s a
Memristor
i(mA)
0.15
0.075
v(V)
-1
- 0.5
0
- 0.075
- 0.15
0.5
1
The HP Memristor
i
+
i (mA)
Pt-2
TiO2-x
v
0
TiO2
̶
v (V)
Pt-1
(a)
2010, Johnson, et al.,
Nanotechnology
(b)
i, mA
2009, R. Waser,
Microelectronic Engineering
i, nA
1000
5
500
v,
0
0
0
mV
0
-100
-1000
-6
100
(d)
2012, Chanthbouala, et al.,
Nature Materials
0
-3
6
3
2013, Nardi et al., IEEE
Trans. Electron Devices
i, μA
i, mA
3
2
100
Reset
Set
1
0
0
-1
Set
-2 Reset
-3
-2 -1 -0.5 0 0.5 1
v,
0
0
V
-100
-2 -1
(e)
V
-500
-5
(c)
v,
0
0
1
2
2000, Beck, et al.,
Applied Physics Letters
(f)
1
20
0
v,
0
V
v,
0
V
-1
-20
-2
-40
0
2
i, mA
40
-0.5
V
2003, Sakamoto, et al.,
Applied Physics Letters
i, μA
0
v,
0.5
-0.2
0
0.2
Weird computer using base-10 instead of
base-2 number ?
Unification of the laws
Ohm’s laws using
fractional derivatives
Fractional derivatives and Fractance
If we consider the set of the three Ohm’relationships, it appears that
they involve: 1 the integral of intensity, 2 the intensity and
3 the derivative of the intensity.
Introducing fractional derivative which
1 t
a kind of interpolation between the
v(t ) 
i ( )d , isorder
of the derivatives we can
C t0
summarize and generalize thoses laws
v(t )  Ri (t ),
in only one,creating a new
mathematical object: the Fractance
d
which represents either a resistor,
v(t )  L i (t )
an inductor or a capacitor
dt

It is possible to generalize the Ohm’s law:
v(t )  an Dtni (t )
Fractional derivatives
The idea of fractional calculus has been known since the development
of the regular calculus and it means a generalization of integration
and differentiation to arbitrary order. There exist several definitions of
the fractional derivatives known since centuries:
(the first example is a letter from Liebniz to the french mathematician
L’Hospital in date of September 30, 1695, about the existence of the
half-order derivative).
They are used for modelling numerous physical systems: dielectric
polarization, visco-elastic systems, electrode-electrolyte
polarization, …
In this presentation we consider both the Riemann-Liouville and the
Caputo’s definition (1967).
We will use also the Grünwald-Letnikov’s definition for numerical
simulations.
The Riemann-Liouville definition
R
a
1
dn
D f (t ) 
(n - ) dt n

t

t
a
(t - )n--1 f ()d 

Using the classical Gamma function
which verifies
( z  1)  z( z )
( z )   e t dt
- t z -1
0
It is possible to write also this definition:
n
d
R 
n -
D
f
(
t
)

j
f (t )  , t  a, n - 1    n
a t
n a t
dt
where
a
jt f (t ) is the integral Riemann-Liouvile operator
t
1

-1
j
f
(
t
)

(
t

)
f ()d 
a t

a
()
Caputo’s definition
Remark: Caputo’s definition as well as Riemann-Liouville’s one are
depending upon a parameter a because fractional derivatives are
non-local and show a memory effect
t
1
n --1 ( n )
D
f
(
t
)

(
t

)
f ()d 
a

a
( n -  )

t
n


d

n -
which can be also written a Dt f (t )  j
 n f (t )  , t  a
 dt

In both definitions ,
n -1    n
Those definitions are equivalent under some conditions.
Fractional derivatives have also special relationships with the Laplace
transform as for example:
n -1
L  0 D f (t )  s L  f (t ) -  s -1- k f ( k ) (0)

t

k 0
Examples of fractional derivative (Caputo’s definition)
f : t  f (t )  t 2 ,
For 1    2
f  : t  f (t )  2t ,
f  : t  f (t )  2
t
2
2(t - a)(2- )
2--1
(t - )
d 
a D f (t ) 

a
(2 - )
(2 - )(2 - )

t
(2 -1)
2(
t
0)
2t
1
  2t  f (t )
0 Dt f (t ) 
(2 - 1)(2 - 1) 1
1,1
0.9
D
f
(
t
)

2.0795(
t
a
)
,
a t
1,3
0.7
D
f
(
t
)

2.2011(
t
a
)
,
a t
1,5
0.5
D
f
(
t
)

2.2568(
t
a
)
,
a t
1,7
0.3
D
f
(
t
)

2.2285(
t
a
)
,
a t
1,9
0.1
D
f
(
t
)

2.1023(
t
a
)
,
a t
1,999
0.001
D
f
(
t
)

2.0012(
t
a
)
 2  f (t )
a t
for
a  0,
Examples of fractional derivative (Caputo’s definition)
f : t  f (t )  t 2 ,
f  : t  f (t )  2t ,
f  : t  f (t )  2
t
2
2(t - a)(1- ) (2a  t - )
1--1
For 0    1 a D f (t ) 
(t - )
d  

a
(1 - )
(2 - )(1 - )(1 - )

t
2(t - 0)(2-1) (2  0  t - 0) 2t 2 2

 t  f (t )
0 D f (t ) 
(2 - 0)(1 - 0)(1 - 0)
2
0
t
0,1
1.9
D
f
(
t
)

1.0945
t
,
0 t
0,3
1.7
D
f
(
t
)

1.2948
t
,
0 t
0,5
1.5
D
f
(
t
)

1.5045
t
,
0 t
0,7
1.3
D
f
(
t
)

1.7024
t
,
0 t
0,9
1.1
D
f
(
t
)

1.9116
t
,
0 t
Fractance (integer exponent)
It is possible to unify the Ohm’s laws setting:
1 t
v(t )   i ( )d ,
C t0
v(t )  Ri (t ),
d
i (t )
dt
1
 C , if

an   R, if
 L, if


D f (t )  
-1
t
-
f ()d 
Dt0 f (t )  f (t )
v(t )  L
and
t
Dt1 f (t ) 
n  -1
n0
n 1
d
f (t )
dt
we obtain, with n = -1, 0,1
v(t )  an Dtni (t )
The term fractance was coined by A. le Mehaute and G. Crepy in 1983.
Fractance (fractional exponent)
If we consider now for
-1    1
the fractional derivatives previously defined. We choose the RiemannLiouville’s definition with a zero origin, that is we consider that
R
0
In order to generalize
we apply the Laplace
transform to both sides
We obtain the impedance
In the case where
Dt  Dt
v(t )  an Dtni (t )
L  v(t )   an s n L i(t )   V (s)  an s n I (s)
V ( s)
Z (s) 
 an s n
I (s)
 of is real this define the fractance Z ( s)  F  s 
We use the Inverse Laplace transform to obtain
v(t )  a Dti (t )
Fractance : actual realization
Electric circuit explicitly built in order to proof the existence
of a fractance element with s = 1/2
Generalized law of Fractance
One can write again the previous equations under the new form:
1 -1
1 0
1
D v(t )  Dt i(t ) or Dt (t )  Dt q(t )
C
C
Dt0v(t )  RDt0i(t ) or Dt1(t )  RDt1q(t )
0
t
Dt-1v(t )  LDt0i(t )
or
Dt0(t )  LDt1q(t )
Which can be represented as a single equation
1 -1
t
v(t )  F
D
1 ,2
 2 -1
t
D
i(t )
or
with
1
t
D (t )  F
1 ,2
2
t
D q(t )
F 1 ,2
C -1 , if 1  1,  2  0,

  R, if 1   2  1,
 L, if   0,   1.
1
2

Generalized law of Fractance
If
1  2  1
if
1  1, 2  0 there is a capacitor,
if
1  0, 2  1 there is an inductor.
the equation stands for a resistor,
For arbitrarily chosen 1 ,  2 between 0 and 1 there is a fractor
device.
All those equations can be represented by the following the general
chart flow
Generalized Fractance (chart flow)
Memfractance
Memristor, memcapacitor, meminductor
A memristor is a resistor which is variable with respect to the time,
depending on the quantity of the electric charge which was passed
through it (since a given initial time) :
instead of R , we have RM (q(t ))  RM (t )
as for example
RM (q (t ))  RM (t )  1  q(t )  q(t ) 2
In 2009, after the discovery of the H.P. Lab. L. O. Chua generalized
the idea of memory element to two other elements: the memcapacitor
and the meminductor
Memristor, memcapacitor, meminductor
A memristor is a resistor which is variable with respect to the time,
depending on the quantity of the electric charge which was passed
through it (since a given initial time) :
RM (t )  RM (q(t ))
In the same way we define a memcapacitor
Example:
-1
M
C (t )  C
and a meminductor
Example:
-1

t
t0
q()d  
LM (t )  LM (q(t ))
LM (t )  LM (q(t ))  1  e q (t )
-1
M
C (t )  C

-1

t
t0
q ( )d 
1
t
1  2   q()d 
t0

2
2nd order Memristor
A 2nd order memristor is a new element which extends the memresistor
principle in order to connect the integral of the flux : (t ) 
to the integral of the charge: (t ) 
We set

t
-
q ()d 
R2 M ((t ))  R2 M (t )
With for example
R2 M ((t ))  1  (t )  (t ) 2
The relationship between both quantities is:
R2 (t )  R2 M ((t ))q(t )
M

t
-
()d 
Memfractance: generalized constitutive relations
We can unify the previous definitions of RM (t ), LM (t ), LM (t ), and R2 M (t ),
in only one:
CM-1 (t ),

 R (t ),
FM1 ,2 (t )   M
 LM (t ),

 R2 M (t ),
if
if
1  1,  2  0,
1   2  1,
if 1  0,  2  1,
if 1  0,  2  0.
The aim: it is possible to proof that Ohm’s laws apply to 1st and 2nd
memristor, meminductor, memcapacitor.
VC (t )  CM-1 (t )q(t ),
We know that
VR (t )  RM (t ) I (t ),
 L (t )  LM (t ) I (t ),
 R2 (t )  R2 M (t )q (t ).
Memfractance (Ohm’s laws)
It is possible to rewrite the previous equations under the form of
relationship betweeen flux and charge:
Dt1C (t )  CM-1 (t ) Dt0 q (t )
Dt1R (t )  RM (t ) Dt1q (t )
Dt0 L (t )  LM (t ) Dt1q (t )
Dt0 R2 (t )  R2 M (t ) Dt0 q (t )
which can be summarized under the form of a single equation
1
t
1 ,2
M
D (t )  F
2
t
(t ) D q(t )
This equation can be generalized in the case where 1 ,  2 are arbitrary
real numbers belonging between 0 and 1.
 ,
We call Memfractance FM 1 2 and Memfractor any electrical device
which exhibits a memfractance.
Memfractance
If
1   2  1 the equation stands for a memristor,
if
1  1, 2  0
it is a memcapacitor,
if
1  0, 2  1
it is a meminductor,
If
1   2  0
it is a 2nd order memristor,
For any
1 ,  2
between 0 and 1 it is a memfractor.
All those equations can be represented by the following the general
chart flow:
Generalized Ohm’s law for memory elements
Generalized Ohm’s law for memory elements
Proposition: The voltage across a memfractance element is given by the
relation


v(t )  Dt1-1 FM1 ,2 (t ) Dt 2 q(t )

which reads also as:
Proposition (Generalized Ohm's Law):
The voltage across a memfractance element can be expressed by the
relation
1-1
t
v(t )  D
F
1 ,2
M
 2 -1
t
(t ) D
i(t )

This Generalized Ohm’s law for memory elements can be proved in the domain
0  1 ,  2  1
Remark 3.3. Ohm's Law in magnetohydrodynamics is also called generalized Ohm's Law [Szabo &
Abonyi, 1965].
We can represent this relation in the following chart flow
Interpolated Memfractance
The fractance which gives the relationship between flux and charge:

Dt1 (t )  FM1 ,2 Dt 2 q(t )
with
1 ,  2
M
F
(t ) 
Dt1 (t )
2
t
D q (t )
2
t
when D q (t )  0
Can be considered as an interpolation of the memresistance, inverse
memcapacitance, meminductance and 2nd-order memresistance,
respectively:
1 ,  2
M
F
1
(t )  a( 1 ,2 )
 b( 1 ,2 ) RM (t )  c( 1 ,2 ) LM (t )  d ( 1 ,2 ) R2 M (t )
CM (t )
Interpolated Memfractance
The coefficients a, b, c, d satisfy the conditions:
Examples of possible coefficients:
1   2
)
2
a( 1 ,2 )  1 (1 -  2 )
b( 1 ,2 )  1 2 (
c( 1 ,2 )   2 (1 - 1 )
d ( 1 ,2 )  (1 - 1 )(1 -  2 )
Interpolated Memfractance
Numerical illustrative examples
We consider the following examples:
RM (q(t ))  1  q(t )  q(t ) 2
LM (t )  1  e
with
R2 M ((t ))  1  (t )  (t ) 2
1
CM ((t )) 
1  (2  (t )) 2
q (t )
i (t )  cos(t ) if t  0,

 i (t )  0 if t  0.
and
(t )  
t
-
q ()d 
which gives the time dependant charge
t
q (t )   i () d   sin t
0
for t  0
We use the Grünwald-Letnikov’s definition, which is more easy to use in
numerical computations.
Voltage versus time
Continuous behavior of memfractance between memristor
and memcapacitor
memcapacitor
memristor
Voltage versus intensity
Voltage versus charge
The flux behavior
Interpolation between memristor and memcapacitor
The flux behavior
Interpolation between memristor and memcapacitor
Memristor 1  1,  2  1
Periodic flux
Memcapacitor 1  1, 2  0
Periodic flux
memfractor
Interpolation between memristor and memcapacitor
1  1,  2  0.1
The flux is divergent
1  1,  2  0.5
The flux is divergent
1  1,  2  0 to 1
Explicit results
2
For the memristor we choose RM (t )  1  q(t )  q (t )
We have
i (t )  cos t
v(t )  FM1 ,2 (t )  Dt2 (sin(t ))
with the approximation
and
q (t )  sin t
(t )  Dt-1  FM1 ,2 (t )  Dt2 (sin(t )) 

D 2 (sin t (t ))  sin(t   2 )
2
In this case the voltage is periodic
 (1   2 )



v(t )  (1 -  2 )  3 - (cos t ) 2   2
1  sin t  (sin t ) 2   sin(t   2 )

2
2


Numerical computations are in good agreement with this approximation.
Explicit results
1  1,  2  0 to 1
However the flux can be increasing or decreasing due to the blue and
red part of the formula.
  22  3 2 - 2 
2 
 2   3( 2 - 1)


cos  2t 
(t )  
 cos  3t 

2
2
2
24






 2    -3 22  39 2 - 42 
 2 ( 2  1) 
 2 
cos

sin  2t 


t 
 
2
4
2
8


 


2
 - 22  3 2  2 
 2 
  2    5 2 _ 48 2  53 
cos

t
cos





 

2
6
2
8








 (  1)   2     2 ( 2  1)
  2 
 2 
sin
1)

3(

cos

sin 
 2 2
2
 t



 
4
8
 2 
 2 
 2  
The flux behavior
The flux is bounded in a band of oscillation
Generalization
A periodic table of circuit elements
Thank you for your memoristable
attention
Brain-like computers
Aplysia with a Nobel Prize Medal
Nobel Prize 2000
Genus
Aplysia:
gastropod
molluscs
called
sea hare or
rabbit marine
Example of Déjà vu Response of the Aplysia
Excitation
Response
Déjà vu response
Déjà vu response is
learning to recognize
and ignore benign
and boring stimulus.
(a) L7
G
Sensory
Neuron
Stimulus
(b)
+
_
v(t)
(c) i(t), Amps
i
+
v
_
20
1
2
3
4
15
5
6
7
8
9
10
10 11
12 13
5
q
0
5
4
3
2
6
7
8 9
12
10 11
14 15
13
8
0
30
40
50
60
70
80
90
750
10
1
2
3
10
20
15
100 110 120 130 140 150
t, seconds
area = 50
5 seconds
Siemens)
Slope = 2 (G = 2 Siemens)
1
20
v(t), Volts
1
Slope =
8
1
(G =
10
14
5
6
7
8
9
10
40
50
60
70
80
90
11 12
13 14
15

0
30
100 110 120 130 140 150
t, seconds
Synapses
are
Memristor!
Hodgkin-Huxley Cells
Alan Hodgkin
Andrew Huxley
1961 Nobel Prize in Physiology
Hodgkin- Huxley Nerve Membrane Model
I
C
+ +
VNa
_
V
INa
_
Time-varying Sodium
conductance
+
VK
GNa _
ENa
IK
GK
GL
EK
EL
Time-varying Potassium
conductance
From
A.L. Hodgkin and A. F. Huxley
A Quantitative Description of Membrane Current and its
Application to Conduction and Excitation in Nerve.
Journal of Physiology, Vol. 117, pp.500-544, 1952
Axon Behaves like a Very large
Inductance
Hodgkin and Cole were shocked to
find the small-signal ac impedance
they had measured from the axon
membrane of squids had a gigantic
inductance (~ 5 Henrys).
Leon Chua
carrying a
1 Henry inductor
The suggestion of an
inductive reactance anywhere
in the system was shocking to
the point of being
unbelievable.
K. S. Cole
Membranes, Ions, and Impulses
University of California
Press , 1972.
Kenneth Cole
Recent researches (2013)
Interpolation entre un memristor et un memcapacitor
Recent researches (2013)
Silicon Neuron (2014)
Memristor working as synapse
(2014)
Memristor working as synapse