RESEARCH - Ricerca sull'intelligenza artificiale specializzata al MIT

RESEARCH

A mathematical model of ephaptic interactions in
neuronal ﬁber pathways: Could there be more
than transmission along the tracts?

Hiba Sheheitli

1 and Viktor K. Jirsa1

1Aix-Marseille University, Inserm, INS UMR_S 1106, Marseille, France

Keywords: Ephaptic interaction modeling, Axonal cable theory, Fiber pathways, Neuronal commu-
nication, White matter, Spatiotemporal patterns

a n o p e n a c c e s s

j o u r n a l

ABSTRACT

While numerous studies of ephaptic interactions have focused on either axons of peripheral
nerves or on cortical structures, no attention has been given to the possibility of ephaptic
interactions in white matter tracts. Inspired by the highly organized, tightly packed geometry
of axons in ﬁber pathways, we aim to investigate the potential effects of ephaptic interactions
along these structures that are resilient to experimental probing. We use axonal cable theory
to derive a minimal model of a sheet of N ephaptically coupled axons. Numerical solutions
of the proposed model are explored as ephaptic coupling is varied. We demonstrate that
ephaptic interactions can lead to local phase locking between adjacent traveling impulses
and that, as coupling is increased, traveling impulses trigger new impulses along adjacent
axons, resulting in ﬁnite size traveling fronts. For strong enough coupling, impulses
propagate laterally and backwards, resulting in complex spatiotemporal patterns. While
common large-scale brain network models often model ﬁber pathways as simple relays of
signals between different brain regions, our work calls for a closer reexamination of the
validity of such a view. The results suggest that in the presence of signiﬁcant ephaptic
interactions, the brain ﬁber tracts can act as a dynamic active medium.

AUTHOR SUMMARY

Starting from the FitzHugh-Nagumo cable model, we derive a system of nonlinear coupled
partial differential equations (PDEs) to model a sheet of N ephaptically coupled axons. Noi
also present a continuous limit approximation transforming the model into a two-dimensional
ﬁeld equation. We numerically solve the equations exploring the dynamics as coupling
strength is varied. We observe phase locking of adjacent impulses and coordination of
subthreshold dynamics. Strong enough coupling generates complex spatiotemporal patterns
as new impulses form traveling fronts propagating laterally and backwards. The transition
between different dynamic regimes happens abruptly at critical values of parameter. IL
results put into question the validity of assuming the role of ﬁber pathways to be that of mere
interneuronal transmission and call for further investigation of the matter.

INTRODUCTION

It has long been thought that signals exchanged between different brain regions are faithfully
transmitted along the white matter tracts through axons that can be modeled as passive electric
cables (Hodgkin & Huxley, 1952). This has led many large-scale network models to assume

Citation: Sheheitli, H., & Jirsa, V. K.
(2020). A mathematical model of
ephaptic interactions in neuronal ﬁber
pathways: Could there be more than
transmission along the tracts?
Network Neuroscience, 4(3), 595–610.
https://doi.org/10.1162/netn_a_00134

DOI:
https://doi.org/10.1162/netn_a_00134

Supporting Information:
https://doi.org/10.1162/netn_a_00134

Received: 8 settembre 2019
Accepted: 26 Febbraio 2020

Corresponding Author:
Hiba Sheheitli
hs497@cornell.edu

Handling Editor:
Mason Porter

Copyright: © 2020
Istituto di Tecnologia del Massachussetts
Pubblicato sotto Creative Commons
Attribuzione 4.0 Internazionale
(CC BY 4.0) licenza

The MIT Press

D
o
w
N
o
UN
D
e
D

F
R
o
M
H

T
T

:
/
/

D
io
R
e
C
T
.

io
T
.

e
D
tu
N
e
N
UN
R
T
io
C
e
–
P
D

F
/

4
3
5
9
5
1
8
6
7
3
2
8
N
e
N
_
UN
_
0
0
1
3
4
P
D

B
sì
G
tu
e
S
T

o
N
0
7
S
e
P
e
M
B
e
R
2
0
2
3

A mathematical model of ephaptic interactions in neuronal ﬁber pathways

Ephaptic interaction:
A form of indirect communication
between cells through the exchange
of ions via the shared extracellular
spazio.

that signals communicated between different brain regions are relayed along the axons of ﬁber
pathways with a ﬁnite speed without any interaction occurring between the traveling signals
along the way (Bassett, Zurn, & Gold, 2018; Breakspear, 2017; Sanz-Leon, Knock, Spiegler, &
Jirsa, 2015). The aim of this work is to motivate a reexamination of this latter highly consequen-
tial assumption. In 1940, Katz and Schmitt investigated the nonsynaptic electrical interaction
between adjacent nerve ﬁbers (Katz & Schmitt, 1940). In their work, two large parallel non-
myelinated axons were isolated from the crab limb nerve. They succeeded in demonstrating
Quello (UN) the passage of an action potential (impulse) in one ﬁber causes subthreshold excitabil-
ity changes in the adjacent ﬁber and (B) when impulses are set up simultaneously along both
ﬁbers, a mutual interaction occurs that can lead to speeding up or slowing down of the impulses
and also possibly to synchronization between the two impulses, depending on the initial phase
relationship. The effect was observed to be ampliﬁed when the resistance of the extracellular
space surrounding the axons was increased. In the next year, a similar study was presented by
Arvanitaki (1942) on giant axons of Sepia ofﬁcinalis (common cuttleﬁsh). In that study, Arvanitaki
coined the term “ephapse” to denote “the locus of contact or close vicinity of two active func-
tional surfaces” (P. 90). The term is derived from the Greek term signifying the act of touching,
as opposed to “synapse” which is derived from the Greek term signifying the act of joining or
linking. Since then, the term ephaptic interaction has been used to refer to communication be-
tween neuronal cells via electrical conduction through the surrounding extracellular space, COME
opposed to communication mediated by chemical synapses or gap-junctions. In 1980, ephap-
tic transmission was observed between spontaneously active single nerve ﬁbers in the spinal
nerve roots of dystrophic mice (Rasminsky, 1980). Shortly after, ephaptic interactions were ob-
served to contribute to neuronal synchrony in rat hippocampal slices (Taylor & Dudek, 1982).
Then in 1984, experiments suggested a role for ephaptic transmission in hemifacial spasm
pathophysiology by causing “cross-talk” between facial nerve ﬁbers (Nielsen, 1984). More
recently, hallmarks of ephaptic interaction were observed in rat cortical pyramidal neurons in
slices, and supported the idea that this interaction facilitates the coordination and possibly the
synchrony of neighboring neurons in the gray matter (Anastassiou, Perin, Markram, & Koch,
2011).
Inoltre, there have been numerous other experimental and modeling investiga-
tions of ephaptic interaction (Anastassiou & Koch, 2015; Barr & Plonsey, 1992; Campana, 1981;
Bokil, Laaris, Blinder, Ennis, & Keller, 2001; Goldwyn & Rinzel, 2016; Grindrod & Sleeman,
1984; Holt & Koch, 1999; Ramón & Moore, 1978; Stacey, Hilbert, & Quail, 2015). Tuttavia,
it can be seen that all these previous studies focused on one of two contexts: (UN) cortical ar-
eas, particularly interactions between neighboring neurons through the resulting local ﬁeld
potential (Anastassiou & Koch, 2015; Anastassiou et al., 2011; Blot & Barbour, 2014; Fröhlich
& McCormick, 2010; Goldwyn & Rinzel, 2016; Holt & Koch, 1999; Taylor & Dudek, 1982)
O (B) peripheral nerves, particularly interactions between myelinated axons in a nerve bun-
dle and inquiries into effects of demyelination (Binczak, Eilbeck, & Scott, 2001; J. W. Clark &
Plonsey, 1970; Marrazzi & Lorente, 1944; Nielsen, 1984; Ramón & Moore, 1978; Rasminsky,
1980; Reutskiy, Rossoni, & Tirozzi, 2003). To our knowledge, there has been no discussion on
ephaptic interaction between axons of the white matter tracts. While the predominant myelina-
tion in white matter axons might be presumed to be preventing ephaptic interference, studies
of myelinated axons in nerves suggest otherwise (Binczak et al., 2001; Marrazzi & Lorente,
1944; Rosenblueth, 1941). Inoltre, some ﬁber pathways can have a considerably high pro-
portion of unmyelinated axons, ad esempio 30% (in regions of the corpus callosum of the adult
rhesus monkey; Lamantia & Rakic, 1990) E 45% (in the splenium of the corpus callosum
of the adult rabbit; Waxman & Swadlow, 1977). It is known that ﬁber pathways in the brain
are constituted of densely packed long axons running in parallel.
In Wedeen et al. (2012),
diffusion magnetic resonance imaging results were presented to illustrate that “cerebral path

Network Neuroscience

596

D
o
w
N
o
UN
D
e
D

F
R
o
M
H

T
T

:
/
/

D
io
R
e
C
T
.

io
T
.

e
D
tu
N
e
N
UN
R
T
io
C
e
–
P
D

F
/

4
3
5
9
5
1
8
6
7
3
2
8
N
e
N
_
UN
_
0
0
1
3
4
P
D

B
sì
G
tu
e
S
T

o
N
0
7
S
e
P
e
M
B
e
R
2
0
2
3

A mathematical model of ephaptic interactions in neuronal ﬁber pathways

crossings formed well-deﬁned 2D sheets” (P. 1631) and that “this sheet structure was found
throughout cerebral white matter and in all species, orientations, and curvatures. Inoltre,
no brain pathways were observed without sheet structure” (Wedeen et al., 2012, P. 1632).
Inoltre, electron micrograph images show that neighboring axons in ﬁber pathways are
often separated by distances as small as 20 nm (Waxman & Swadlow, 1977), which would sug-
gest a relatively high extracellular space resistance. These latter geometric characteristics set
favorable conditions for ephaptic exchanges to be at play in white matter ﬁber pathways. Ide-
alleato, direct experimental examination of the activity of axons in the white matter would serve
to accurately quantify ephaptic interactions there. Tuttavia, probing into the inner workings
of the white matter remains a challenging endeavor, mainly because of technical limitations
on the temporal and spatial resolution of current noninvasive imaging techniques (Lui, Yang,
Wilke, & Yuan, 2011). Inspired by these facts, we wish to investigate the matter by putting
forward a simple but realistic mathematical model of excitable axons arranged in a sheetlike
geometry and coupled through a resistive extracellular space.

In the Materials and Methods, we start from local circuit theory and the cable model of an
axon to derive a model for a sheet of N ephaptically coupled axons. We then make a contin-
uous limit approximation to transform the resulting model of N coupled 1D partial differential
equations (PDEs) into a 2D PDE that can be seen as a ﬁeld equation governing the dynamics
of a sheet of coupled axons. In the Results section, we numerically solve the equations and
explore the different possible dynamical regimes along with examining the equivalence of the
two proposed models. In the Discussion section, we discuss the potential ramiﬁcations of the
results along with future work directions that this work motivates.

MATERIALS AND METHODS

The Mathematical Model

Our goal here is to put forward a minimal model that possesses the key elements that allow
the study of the effects of ephaptic interactions on action potential transmission along ﬁber
pathways. Given the densely packed parallel geometric arrangement of axons in the white
matter, we assume that currents generated during action potential propagation are mainly
axial in direction, both inside the axons and in the surrounding extracellular space ( J. Clark
& Plonsey, 1968; Plonsey, 1977). Then the axons can be represented by what is known as
the cable equation, while the extracellular space between axons can be represented by an
effective longitudinal resistance per unit length (Rall, 1962; Scott, 2002). Such a model for two
ephaptically coupled axons is derived in Bell (1981). Figura 1 depicts the equivalent circuit
model used to derive the cable equations for two ephaptically coupled axons. The following
notation is used here:

ia: axial (axoplasmic) current inside the axon per unit length
ie: axial current in the extracellular space surrounding the axons per unit length
io sono: axonal transmembrane current per unit area
ra: axoplasmic resistance per unit length
Rif: extracellular space resistance per unit length
C: axonal membrane capacitance per unit length
jion: active ionic current ﬂowing across the axonal membrane per unit length
G: membrane conductance per unit length
z: distance along the axon
vm: transmembrane potential of an axon

Cable equation:
A partial differential equation
describing the evolution of cell
membrane voltage and currents as
functions of distance and time.

Network Neuroscience

597

D
o
w
N
o
UN
D
e
D

F
R
o
M
H

T
T

:
/
/

D
io
R
e
C
T
.

io
T
.

e
D
tu
N
e
N
UN
R
T
io
C
e
–
P
D

F
/

4
3
5
9
5
1
8
6
7
3
2
8
N
e
N
_
UN
_
0
0
1
3
4
P
D

B
sì
G
tu
e
S
T

o
N
0
7
S
e
P
e
M
B
e
R
2
0
2
3

A mathematical model of ephaptic interactions in neuronal ﬁber pathways

Figura 1. Schematic of the equivalent circuit model for two ephaptically coupled axons.

va: axoplasmic potential inside an axon
ve: electric potential in the extracellular space
IO: external applied current per unit length
UN, B: parameters of the FitzHugh-Nagumo model

Here, all currents and potential variables are varying functions of time and axial location z.

Kirchoff’s law:
The law of conservation of electric
charge.

In the limit of ∆z

0, Kirchoff’s current law gives the following relationships between the
transmembrane, axial, and extracellular currents; subscripts 1 E 2 each refer to one of the
two identical axons:

→

io sono
1 =

−

io sono
2 =

∂ia
2
∂z

;

−

= im

1 + io sono
2 .

∂ia
1
∂z
∂ie
∂z

Ohm’s law:
The law of proportionality between
the electric current and voltage
across a conductor.

Inoltre, Ohm’s law relates the currents to the electric potentials as follows:

∂va
1
∂z

raia

1 ,

−

∂va
2
∂z

raia
2

−

∂ve
∂z

reie.

−

Inoltre, the transmembrane current for each axon can be expressed as

im = c

∂vm
∂t

+ jion −

IO, where vm = va

ve.

−

(1)

(2)

(3)

(4)

The term jion represents the active transmembrane currents due to ion channel activity that is
nonlinearly dependent on the transmembrane voltage. Detailed mathematical representation
of the dependence of jion on vm was described in the seminal work by Hodgkin and Huxley
(1952), which utilized three variables to represent the kinetics of ion channel activation. In
1961, FitzHugh proposed a simpliﬁcation of that model that utilizes only one recovery variable
(FitzHugh, 1961):

jion = gf (vm, w) , where f(v, w) =

−

− (cid:18)

v3
3 −

(cid:19)

(5)

con

∂w
∂t

= ε (v + UN

bw) .

−

Here, w is a slow recovery variable. To arrive at the cable equation model, we need to combine
all the above relationships to eliminate the current variables. We start by differentiating the

Network Neuroscience

598

D
o
w
N
o
UN
D
e
D

F
R
o
M
H

T
T

:
/
/

D
io
R
e
C
T
.

io
T
.

e
D
tu
N
e
N
UN
R
T
io
C
e
–
P
D

F
/

4
3
5
9
5
1
8
6
7
3
2
8
N
e
N
_
UN
_
0
0
1
3
4
P
D

B
sì
G
tu
e
S
T

o
N
0
7
S
e
P
e
M
B
e
R
2
0
2
3

A mathematical model of ephaptic interactions in neuronal ﬁber pathways

expression for vm with respect to z and substituting Equation 3 in it:

∂vm
1
∂z

−

raia

1 + reie

∂vm
2
∂z

−

raia

2 + reie.

Differentiating again and substituting Equation 1, we obtain the following:

∂2vm
1
∂z2 = (ra + Rif) io sono
∂2vm
2
∂z2 = (ra + Rif) io sono

1 + reim
2 ,

2 + reim
1 .

(6)

D
o
w
N
o
UN
D
e
D

F
R
o
M
H

T
T

:
/
/

D
io
R
e
C
T
.

io
T
.

e
D
tu
N
e
N
UN
R
T
io
C
e
–
P
D

F
/

4
3
5
9
5
1
8
6
7
3
2
8
N
e
N
_
UN
_
0
0
1
3
4
P
D

B
sì
G
tu
e
S
T

o
N
0
7
S
e
P
e
M
B
e
R
2
0
2
3

Solving the above system of two equations for an expression for im
2 , then combining the
result with Equation 4, we arrive at the cable equations for two ephaptically coupled axons:

1 and im

∂2vm
1
∂z2 −
∂2vm
2
∂z2 −

∂2vm
2
∂z2 = c
∂2vm
1
∂z2 = c

∂vm
1
∂t

∂vm
2
∂t

+ jion,1

+ jion,2

I1,

I2,

−

where γ =

ra + Rif
2rare + r2

UN (cid:19)

(cid:18)

, α =

Rif
2rare + r2

UN (cid:19)

(cid:18)

(7)

We can see that for zero extracellular resistance, the two cable equations are uncoupled such
that any current exiting one axon will immediately dissipate in the extracellular space and no
exchange between the axons can occur. The resulting single FitzHugh-Nagumo cable was ﬁrst
put forward by Nagumo in Nagumo, Arimoto, and Yoshizawa (1962). It can be seen that the
cable equation is the classical 1D diffusion equation with an added term, jion. The presence of
the nonlinear active currents renders the cable excitable, such that, if the membrane potential
is perturbed from its resting value, it will return to that value unless the perturbation is strong
enough to elicit the large action potential response that will then be propagated along the
axon, away from the location of perturbation, with the signal’s shape preserved.

We wish to extend the model to a sheet of ephaptically coupled axons, questo è, an N number
of axons coupled through the extracellular space. Figura 2 shows a schematic of a cross-
sectional view of such an arrangement where we represent the cross sections of axonal cables
as nodes on a line, interspaced with nodes representing extracellular space.

A model of such a conﬁguration of N number of coupled axons was presented in Grindrod
and Sleeman (1984). The main assumption made in this latter work is that each axon is only
coupled to the two axons that are positioned directly next to it. While this later assumption
is common in network models, the authors offered no physiological justiﬁcation for it in the
context of ephaptically coupled axons. Invece, we will start from the more physical assump-
tion that transmembrane currents are radially uniform, such that we can reasonably consider
im for each cable to be equally partitioned into two parts feeding into the extracellular space
(represented as nodes) adjacent to it. Consequently, while the previously presented model

Figura 2. Schematic of a cross-sectional view of the sheet of N axons model.

Network Neuroscience

599

A mathematical model of ephaptic interactions in neuronal ﬁber pathways

restricts ephaptic interactions to nearest neighbor axons (Grindrod & Sleeman, 1984), our
model allows each axon to interact with all other axons through the shared extracellular space
with the coupling strength decaying with distance between interacting axons. To arrive at that,
we take the potential in the extracellular space for an axon positioned at a node q to be the
average of the potential at its adjacent extracellular nodes such that

q = va
vm

q −

1
2 (cid:16)

ve
q

−

1 + ve

q+1

(cid:17)

(8)

Note that the index q refers to the node number on the line, so for N axons, q takes values
between 1 and 2N + 1, such that va
q on nodes
q = 1, 3, . . . , N + 1. Differentiating Equation 8 with respect to z, and making use of Ohm’s
law, we obtain

q is deﬁned on nodes q = 2, 4, . . . , N and ve

∂vm
q
∂z

raia

q +

−

1
2

Rif

ie
q

−

(cid:16)

1 + ie

q+1

(cid:17)

(9)

From Kirchoff’s ﬁrst law of current, we have the following:

∂ia
q
∂z

io sono
q ,

−

∂ie
q
−
∂z
∂ie

q+1
∂z

1
2 (cid:16)
1
2 (cid:16)

2 + io sono
q

io sono
q
−

io sono
q+2 + io sono
q

(cid:17)

These latter equations are a generalization of Equations 1 E 2. Differentiating Equation 9

again and plugging in the above current relationships, we arrive at

∂2vm
q
∂z2 =

ra +

(cid:18)

1
2

Rif(cid:19)

io sono
q +

1
4

Rif

io sono
q
−

(cid:16)

2 + io sono

q+2

(cid:17)

(10)

For the cables that are at the two ends of the line, the corresponding relationship would be

the following:

∂2vm
2
∂z2 =

ra +

(cid:18)

∂2vm
N
∂z2 =

ra +

(cid:18)

1
2

io sono
2 +

Rif(cid:19)

1
4

Rif (io sono

4 ) ,

io sono
N +

Rif(cid:19)

1
4

Rif

io sono
N

−

(cid:1)

(cid:0)

We have obtained a system of N equations relating im and vm of all the axons. This is the
equivalent of Equation 6 for the two-axon system. The linear system of N equations can be
solved, such that we can express each im explicitly in terms of vm of all the axons. The solution
takes the form

io sono
p =

4
Rif

N
∑
s=1

αps

∂2vm
S
∂z2

;

(11)

p = 1, 2, . . . , N referring to the N axons. The α’s represent coupling strength between each
, where A is

pair of axons and are obtained as the elements of the inverse matrix A−
the tridiagonal matrix:

1 =

αij

(cid:2)

(cid:3)

600

Network Neuroscience

D
o
w
N
o
UN
D
e
D

F
R
o
M
H

T
T

:
/
/

D
io
R
e
C
T
.

io
T
.

e
D
tu
N
e
N
UN
R
T
io
C
e
–
P
D

F
/

4
3
5
9
5
1
8
6
7
3
2
8
N
e
N
_
UN
_
0
0
1
3
4
P
D

B
sì
G
tu
e
S
T

o
N
0
7
S
e
P
e
M
B
e
R
2
0
2
3

A mathematical model of ephaptic interactions in neuronal ﬁber pathways

D 1
1 D 1
1 D
. . .

A =




















1
. . .
. . .

. . .
. . .
1




















. . .
D 1
1 D 1
1 D

with D = 4

R +

(cid:18)

1
2 (cid:19)

and R =

ra
Rif

Explicit algebraic expressions for the elements of the inverse of such a tridiagonal matrix are
presented in Usmani (1987) and indicate that the ephaptic effect increases as the ratio R de-
creases and that α for two axons on the line decreases as the distance between them increases.
Tuttavia, for the numerical solutions presented in the sections to follow, we found it simpler
to numerically compute the inverse of A instead of the individual α′s.

Combining Equations 11, 5, E 4, we obtain the model for a sheet of N ephaptically

coupled FitzHugh-Nagumo cables:

D
o
w
N
o
UN
D
e
D

F
R
o
M
H

T
T

:
/
/

D
io
R
e
C
T
.

io
T
.

4
Rif

N
∑
s=1

αps

∂2vm
S
∂z2 = c

∂vm
P
∂t

+ gf

vm
P , wp

Ip,

(cid:17) −

(cid:16)

(12)

∂wp
∂t

vm
P + UN

= ε

(cid:16)

bwp

(cid:17)

−

We nondimensionalize the space and time variables as follows:

˜z =

G (ra + Rif)

(cid:19)

(cid:18)q

z ,

˜t =

G
C

The system becomes

e
D
tu
N
e
N
UN
R
T
io
C
e
–
P
D

F
/

4
3
5
9
5
1
8
6
7
3
2
8
N
e
N
_
UN
_
0
0
1
3
4
P
D

4 (R + 1)

N
∑
s=1

αps

∂2vm
S
∂ ˜z2 =

∂vm
P
∂˜t

+ F

vm
P

(cid:16)

(cid:17) −

ˆIp,

∂wp
∂˜t

vm
P + UN

= ˆε

(cid:16)

bwp

(cid:17)

−

where ˆIp =

IO
G

ˆε = ε

C
G

B
sì
G
tu
e
S
T

o
N
0
7
S
e
P
e
M
B
e
R
2
0
2
3

From now on, we drop the superscript of the transmembrane voltage along with the tilde and
hat. The ﬁnal equations take the following form:

4 (R + 1)

N
∑
s=1

αps

∂2vs
∂z2 =

∂vp
∂t

+ F

vp, wp

(cid:0)

Ip,

−

(cid:1)

(13)

∂wp
∂t

= ε

vp + UN

(cid:0)

−

bwp

(cid:1)

For the results that will follow, we choose the following values for the FitzHugh-Nagumo re-
covery variable: a = 0.7, b = 0.5, and ǫ = 0.1. Then, we are left with one free parameter R
that reﬂects the strength of the ephaptic interaction. The goal is to investigate the dynamics
of the system as this parameter is varied. We note here that the resulting emergent network

Network Neuroscience

601

A mathematical model of ephaptic interactions in neuronal ﬁber pathways

behavior is robust against variations of the FitzHugh-Nagumo parameters within a reasonable
range that preserves the relevant dynamic features of the neuron model (planar, class 1 in terms
of excitability, existence of refractory period, and slow manifold). Multiple parameters were
tried, and only one example is presented here for brevity.

Estimation of the Coupling Strength Parameter

To estimate the physically plausible range of values for parameter R, we start from the deﬁnition

R =

ra
Rif

1
ρ

Ae
Aa

where Ae and Aa are the cross-sectional areas of the extracellular space and the axon, respec-
tively. ρ is the ratio of extracellular to intracellular resistivity, and is typically assumed to be in
the range of 1 A 4 (Goldwyn & Rinzel, 2016). Given that axons of the ﬁber pathways are very
tightly packed, we consider that the cross-sectional area of the space between adjacent axons
can range from a tenth to several multiples of the cross-sectional area of the axon. Based on
Quello, we consider R to vary between 0 E 1.

Continuous Limit Approximation

Our model is a system of N coupled nonlinear PDEs, each representing one distinct cable
and accompanied by an ordinary differential equation for the corresponding slow recovery
variable. Tuttavia, neighboring axons in ﬁber pathways are often very densely packed, come
as the distances separating two adjacent axons are considerably small relative to the axonal
diameter (Waxman & Swadlow, 1977). For this reason, we will make the approximation that
the variables vm and im, while being only physically deﬁned for the axonal space, can be
abstractly represented by continuous ﬁeld variables v and i. If we go back to Equation 10, we
notice that the last two terms can be rewritten by using the following discrete approximation
of a second partial derivative (Abramowitz & Stegun, 2012):

∂2im
∂x2 ≈

io sono
q
−

2 −

Poi

q + io sono

q+2

2io sono
4δ2

io sono
q
−

2 + io sono
q
−

2 ≈

4δ2 ∂2im

∂x2 + 2io sono
q ,

where δ is the small separation between adjacent axons. Using this latter ﬁnite difference

stencil, Equazione 10 is transformed to

∂2vm
∂z2 = (ra + Rif) io sono + δ2re

∂2im
∂x2 .

(14)

This equation relates the transmembrane voltage and current approximate ﬁeld variables.
The second relationship between the two is given by the balance of currents Equation 4. IL
approximate continuous system then takes the following form:

∂2v
∂z2 = (ra + Rif) io + δ2re

∂2i
∂x2

i = c

∂v
∂t

+ gf (v, w)

IO,

−

con

∂w
∂t

= ε (v + UN

bw) ,

−

Network Neuroscience

602

D
o
w
N
o
UN
D
e
D

F
R
o
M
H

T
T

:
/
/

D
io
R
e
C
T
.

io
T
.

e
D
tu
N
e
N
UN
R
T
io
C
e
–
P
D

F
/

4
3
5
9
5
1
8
6
7
3
2
8
N
e
N
_
UN
_
0
0
1
3
4
P
D

B
sì
G
tu
e
S
T

o
N
0
7
S
e
P
e
M
B
e
R
2
0
2
3

A mathematical model of ephaptic interactions in neuronal ﬁber pathways

where the superscripts were dropped for brevity. Prossimo, we nondimensionalize the equations
using the following rescaling:

˜z =

G (ra + Rif)

(cid:19)

(cid:18)q

z ,

˜t =

G
C

˜x =

X
λ

Here, λ is a characteristic lateral length scale of the same order of magnitude as the average
axonal diameter (µm). The resulting system becomes

∂2v
∂ ˜z2 = i + K
∂w
∂˜t

= ˆε (v + UN

∂2i
∂ ˜x2

i =

∂v
∂˜t

+ F (v, w)

ˆI;

−

bw)

, with K =

−

δ2
λ2

1
R + 1

Dropping the tilde and hat for brevity, we obtain the ﬁnal form of the approximate continuous
ﬁeld equations for a sheet of ephaptically coupled axons:

∂2i
∂x2

i =

∂v
∂t

+ F (v, w)

IO;

−

(15)

∂2v
∂z2 = i + K
∂w
∂t

= ε (v + UN

bw) .

−

Green’s function:
An integral transform that aids
in reformulating and solving
differential equations.

We note here that the latter equations governing v and i can be transformed into a more
compact form of one partial integro-differential equation using Green’s function and contour
integration, such that the system becomes the following:

∂v (X, z, T)
∂t

−

F (v, w) + IO +

X, x′

(cid:0)

(cid:1)

∂2v (x′, z, T)
∂z2

dx′,

∂w (X, z, T)
∂t

= ε (v + UN

bw) ,

−

where g

X, x′

= g

(cid:0)
In the results that follow, we take δ2

(cid:0)

(cid:1)

−

x′

(cid:1)

1
√K

sin

(cid:18)

1
√K (cid:0)

x′

−

(cid:19)

(cid:1)

0.01, to be consistent with the assumption that the
interaxonal spacing is very small compared with the characteristic length. Hence, we consider
values of K in the range of 0 A 0.1 to be in line with the above choice of R being between 0
E 1.

λ2 ≈

We also note that the model can be extended to the 3D case by considering axons on a
two-dimensional grid (X, sì) instead of a 1D line (X). While deriving the N coupled PDE system
will be more tedious in this case, the continuous limit approximation leads to Equation 15 con
one added term on the right-hand side of the ﬁrst equation:

∂2v
∂z2 = i + K

∂2i
∂x2 +

∂2i
∂y2 (cid:19)

(cid:18)

i =

∂v
∂t

+ F (v, w)

IO;

−

(16)

∂w
∂t

= ε (v + UN

bw) .

−

Numerical Implementation

Numerical solutions of the two model systems were obtained using the Crank-Nicolson ﬁ-
nite difference method. After numerical experiments were performed with smaller values and

Network Neuroscience

603

D
o
w
N
o
UN
D
e
D

F
R
o
M
H

T
T

:
/
/

D
io
R
e
C
T
.

io
T
.

e
D
tu
N
e
N
UN
R
T
io
C
e
–
P
D

F
/

4
3
5
9
5
1
8
6
7
3
2
8
N
e
N
_
UN
_
0
0
1
3
4
P
D

B
sì
G
tu
e
S
T

o
N
0
7
S
e
P
e
M
B
e
R
2
0
2
3

A mathematical model of ephaptic interactions in neuronal ﬁber pathways

conﬁdence in the stability of the solution was established, we chose a time step of 0.05 and a
spatial step of 0.5 E 1 for the z and x directions, rispettivamente. Zero ﬂux boundary conditions
were enforced, such that the ﬁrst spatial derivative of v remains 0 at the boundaries for all time.
To investigate the response of the system, impulses were initiated at the inlet of axons using
a brief and localized input current I = 2 applied to the axon for t
[0, 4]. Tutto
numerical computation was implemented using Python.

[0, 2] and z

∈

RESULTS

Numerical simulations were performed to investigate the dynamics of the system as the ephap-
tic interaction strength was varied. Inoltre, we compared the dynamics of the continuous
limit system (Equazione 15) to that of the original discrete model (Equazione 13).

Phase Interactions

In Figure 3, two adjacent axons are stimulated such that an action potential is initiated along
each of them and travels from left to right. The timing of the stimulation is such that one
impulse lags behind the other. The shape of the action potentials and their propagation along
the z-direction can be seen in the Supporting Information (SI Figure 1). In Figure 3A, dove noi
set R = 0.8 corresponding to weak ephaptic coupling, the two impulses travel independently
without inﬂuencing each other.
In Figure 3B and C, for the same value of R but with the
stimulated axons directly adjacent to each other, ephaptic interaction causes the two impulses
to attract/repel each other such that the lag between them decreases/increases, after which
they remain locked together, depending on the initial time lag between them. This type of
interaction has been experimentally observed in Katz and Schmitt (1940).

D
o
w
N
o
UN
D
e
D

F
R
o
M
H

T
T

:
/
/

D
io
R
e
C
T
.

io
T
.

e
D
tu
N
e
N
UN
R
T
io
C
e
–
P
D

F
/

4
3
5
9
5
1
8
6
7
3
2
8
N
e
N
_
UN
_
0
0
1
3
4
P
D

B
sì
G
tu
e
S
T

o
N
0
7
S
e
P
e
M
B
e
R
2
0
2
3

Figura 3. Numerical simulation of Equation 13. The color bar indicates the value of v, the x
variable indicates the axon number. Each column corresponds to a simulation with a speciﬁc value
of the parameter; snapshots show progress of time from top to bottom. (UN) R = 0.8, axons number
30 E 20 are stimulated at t = 0 and t = 10, rispettivamente, and the panel rows from top to bottom
correspond to t = 500, 1100, 1700, 2300, 2900. (B) Same as in A but with axons number 25 E
24 stimulated. (C) Same as in B but with stimulation at t = 0 and t = 11. (D) same as in A but with
R = 0.4, and panels show t = 500, 1400, 2300, 3200, 4100.

Network Neuroscience

604

A mathematical model of ephaptic interactions in neuronal ﬁber pathways

Spatial Patterns Generation

Prossimo, we increase the coupling strength by decreasing R, and observe that a transition occurs
where each traveling impulse triggers new action potentials in its two immediately adjacent
axons and the three neighboring impulses move together as a ﬁnite size traveling front, COME
shown in Figure 3D. Further increase in coupling strength leads to the next two adjacent axons
being activated (Figure 4A). Because of the presence of the scaling factor δ2
λ2 in the expression
for K, we do not expect Equation 15 to be equivalent to Equation 13 for the same values
of R. Nonetheless, it can be seen in Supporting Information Figure 2 that the same behavior
described so far also occurs in Equation 15 as the coupling strength K is increased. Inoltre,
as coupling strength is further increased, more and more impulses are triggered as the traveling
front of impulses diffuses laterally and widens (Figure 5A and B). Even further increase in

D
o
w
N
o
UN
D
e
D

F
R
o
M
H

T
T

:
/
/

D
io
R
e
C
T
.

io
T
.

e
D
tu
N
e
N
UN
R
T
io
C
e
–
P
D

F
/

4
3
5
9
5
1
8
6
7
3
2
8
N
e
N
_
UN
_
0
0
1
3
4
P
D

B
sì
G
tu
e
S
T

o
N
0
7
S
e
P
e
M
B
e
R
2
0
2
3

Figura 4.
as in Figure 3D but with R = 0.19 and R = 0.15, rispettivamente.

(UN, B) same as in Figure 3A but with R = 0.33 and R = 0.191, rispettivamente. (C, D) same

Figura 5. Same as in Figure 3 but for Equation 15, with different values of K and for t = 500,
1300, 2200, 3100, 3999. (UN) K = 0.0255. (B) K = 0.026. (C) K = 0.038. (D) K = 0.04.

Network Neuroscience

605

A mathematical model of ephaptic interactions in neuronal ﬁber pathways

K (Figure 5C and D) leads to new fronts of impulses being induced in the forward but also
backward direction, resulting in dynamic spatiotemporal patterns.

Cosine similarity:
The cosine of the angle between two
vettori; computed as the dot product
between the two vectors divided by
the product of their magnitudes.

Poisson process:
A series of independent stochastically
occurring events with a constant
average time interval between events.

Figure 4B, C, and D show that the same transitions also occur in Equation 13 as R is de-
creased, albeit the resulting patterns are more discrete and irregular. To better compare the
responses of the two systems in this regime, we compare the discrete Fourier transform of the
spatial patterns of the two systems at several time instants (Supporting Information Figure 3).
It can be seen that the spatial modal decomposition of the two is rather close, as quantiﬁed
by the cosine similarity between the discrete Fourier transforms at every time step (Supporting
0.006, which indicates close
Information Figure 4). The mean value over time was
similarity. Tuttavia, for Equation 15, these spatiotemporal patterns persist only for a small
range of the parameter, as a further increase in K causes the system to revert back to the later-
ally diffusing traveling front (Figura 6, right column). This is unlike Equation 13 in which the
complex spatio-temporal patterns persist as R is further decreased (Figura 6, left column).

0.96

≈

Spike Trains Interactions

Prossimo, we stimulate all of the 50 axons and observe the collective dynamics. Each axon is
stimulated by a ﬁnite train of 10 impulses. The intervals between impulses are generated by
(The mean interval is the mean value of
a Poisson process with a speciﬁed mean interval.
the intervals between consecutive impulses.) The evolution of the resulting action potentials
for the case of negligible ephaptic coupling is shown in Figure 7A. When ephaptic coupling
becomes signiﬁcant, as in Figure 7B, the impulses self-organize into phase-locked traveling
fronts. A similar effect also occurs for Equation 15 (compare Figure 7C and D).

It has been stated that “in theory, whereas the pattern of Poisson-like impulse codes was
modiﬁed during long-distance propagation, their mean rate was conserved” (Moradmand &
Goldﬁnger, 1995, P. 2415). On the contrary, here in the presence of ephaptic interactions, IL
mean interspike interval (mISI), which is the inverse of the mean rate, decreases with increasing
z location (Supporting Information Figure 5 and Supporting Information Figure 6). The effect is
clearly seen when the mISI is averaged over the 50 axons at downstream z locations (Figura 8).

Figura 6. Numerical simulation of Equation 13 for R = 0.05 (left) and of Equation 15 for K = 0.05
(right). The panels from top to bottom correspond to t = 500, 1100, 1700, 2300, 2900.

Network Neuroscience

606

D
o
w
N
o
UN
D
e
D

F
R
o
M
H

T
T

:
/
/

D
io
R
e
C
T
.

io
T
.

e
D
tu
N
e
N
UN
R
T
io
C
e
–
P
D

F
/

4
3
5
9
5
1
8
6
7
3
2
8
N
e
N
_
UN
_
0
0
1
3
4
P
D

B
sì
G
tu
e
S
T

o
N
0
7
S
e
P
e
M
B
e
R
2
0
2
3

A mathematical model of ephaptic interactions in neuronal ﬁber pathways

Figura 7. Numerical simulation of Equation 13 (UN, B) and Equation 15 (C, D). spike trains of an
average of 10 impulses are triggered along each axon, with a mean interimpulse interval of 10. IL
panels from top to bottom correspond to t = 2499, 4999, 7499, 9999.

D
o
w
N
o
UN
D
e
D

F
R
o
M
H

T
T

:
/
/

D
io
R
e
C
T
.

io
T
.

e
D
tu
N
e
N
UN
R
T
io
C
e
–
P
D

F
/

4
3
5
9
5
1
8
6
7
3
2
8
N
e
N
_
UN
_
0
0
1
3
4
P
D

B
sì
G
tu
e
S
T

o
N
0
7
S
e
P
e
M
B
e
R
2
0
2
3

Figura 8. Mean interspike interval (mISI) averaged over the 50 axons for downstream z locations
for the simulations in Figure 7A (red), 7B (light red), 7C (blue), and 7D (light blue) (low coupling
in dark colors and high coupling in light colors).

Note that Figure 7 shows only a portion of the simulation; in order to compute the mISI values
shown in Supporting Information Figure 5 and Supporting Information Figure 6, we let the
simulation run long enough until all the impulses that were initiated at one end of the axon
reach the other end, such that the total simulation time was 18,000.

DISCUSSION

We presented a minimal model for a sheet or volume of ephaptically coupled axons and
explored its dynamics for a physically plausible range of parameters. We found that the
model captures the experimentally observed attraction/repulsion effect between neighboring

Network Neuroscience

607

A mathematical model of ephaptic interactions in neuronal ﬁber pathways

impulses. For strong enough coupling, the model predicts that action potentials traveling down
an axon can trigger new action potentials in adjacent axons to be initiated and carried along
with it, forming a ﬁnite size traveling front. These fronts increase in size as more axons are
recruited at higher coupling strength. Simulations with even higher coupling strength result
in recurrence of impulses and backward propagation such that a pair of individual impulses
initiated on two nonadjacent axons evolve into trains of impulses that diffuse laterally in the x
direction as well as in both +ve and -ve z directions along the axons. We have also observed
that ephaptic coupling can lead to self-organization among trains of impulses and signiﬁcant
alteration in the timing of action potentials, which is known to be a key element in neuronal
coding (Debanne, 2004). This suggests that ephaptic interactions along ﬁber pathways can
theoretically play an active role in neuronal signal processing in the brain. The numerical
simulations showed that the continuous limit approximation system mimics the qualitative be-
havior of the original model for a speciﬁc range of parameters. This continuous limit model
offers the advantage of being mathematically more compact, more analytically tractable, E
less numerically expensive to solve, and it allows for easy extension of the model to full 3D
geometry.
It furthermore allows for a more intuitive interpretation of the ephaptic coupling
terms and, in its integro-differential form, makes it intuitive that the ephaptic coupling creates
a modulation of the diffusion in the axial direction with an alternating positive and negative
diffusion term on a spatial length scale favoring structures on the scale of √K.

In conclusion, we propose that the various nontrivial responses observed in our numeri-
cal exploration of ephaptic interaction might play an important and complex active role in
interarea neuronal signal transmission and processing in the brain. We hope these results will
motivate a critical examination of the validity of the common assumption that neuronal ﬁber
pathways merely act as transmission cables relaying signals between different brain regions.
In contrast to that latter viewpoint, this theoretical investigation suggests that ephaptic inter-
actions enhanced by the orientations and bundling of neuronal tracts in three-dimensional
space can render the ﬁber pathways an active axonal medium that can give rise to complex
spatiotemporal dynamics.
If this emergent dynamics occurs under physiologically realistic
conditions, then it would be a major so far unknown contributor to information processing
in neural systems. We see various directions that this work can take in the future, including
further exploration of the rich repertoire of responses for different types of stimuli, account-
ing for variability in axonal diameters that will add spatial heterogeneity in the parameters.
Inoltre, a natural extension to a more biophysically detailed treatment of the matter is
possible through incorporating the effect of myelination into the cable equation used in the
modello. Finalmente, we hope that our work will inspire experimental work that can provide precise
quantiﬁcation and characterization of the elusive effects of ephaptic interactions in the axonal
tracts of the brain.

SUPPORTING INFORMATION

Supporting information for this article is available at https://doi.org/10.1162/netn_a_00134.

AUTHOR CONTRIBUTIONS

Hiba Sheheitli: Conceptualization; Data curation; Formal analysis; Investigation; Methodol-
ogy; Validation; Visualization; Writing – Original Draft; Writing – Review & Editing. Viktor K.
Jirsa: Conceptualization; Formal analysis; Funding acquisition; Investigation; Methodology;
Project administration; Resources; Supervision; Writing – Review & Editing.

Network Neuroscience

608

D
o
w
N
o
UN
D
e
D

F
R
o
M
H

T
T

:
/
/

D
io
R
e
C
T
.

io
T
.

e
D
tu
N
e
N
UN
R
T
io
C
e
–
P
D

F
/

4
3
5
9
5
1
8
6
7
3
2
8
N
e
N
_
UN
_
0
0
1
3
4
P
D

B
sì
G
tu
e
S
T

o
N
0
7
S
e
P
e
M
B
e
R
2
0
2
3

A mathematical model of ephaptic interactions in neuronal ﬁber pathways

FUNDING INFORMATION

Viktor K. Jirsa, European Commission (http://dx.doi.org/10.13039/501100000780), Award ID:
H2020-720270.

REFERENCES

Abramowitz, M., & Stegun, IO.

(2012). Handbook of mathemati-
cal functions: With formulas, graphs, and mathematical tables.
New York, NY: Dover Publications.

Anastassiou, C. A., & Koch, C.

(2015). Ephaptic coupling to en-
dogenous electric ﬁeld activity: Why bother? Curent Opinion in
Neurobiology, 31, 95–103.

Anastassiou, C. A., Perin, R., Markram, H., & Koch, C.

(2011).
Ephaptic coupling of cortical neurons. Nature Neuroscience, 14,
217–223.

Arvanitaki, UN. (1942). Effects evoked in an axon by the activity of a
contiguous one. Journal of Neurophysiology, 5(2), 89–108.

Barr, R. C., & Plonsey, R.

(1992). Electrophysiological interac-
tion through the interstitial space between adjacent unmyeli-
nated parallel ﬁbers. Biophysical Journal, 61(5), 1164–1175.
Bassett, D. S., Zurn, P., & Gold, J. IO. (2018). On the nature and use of
models in network neuroscience. Nature Reviews Neuroscience,
19, 566–578.

Campana, J. (1981). Modeling parallel, unmyelinated axons: Pulse trap-
ping and ephaptic transmission. SIAM Journal on Applied Math-
ematics, 41(1), 168–180.

Binczak, S., Eilbeck, J. C., & Scott, UN. C. (2001). Ephaptic coupling
of myelinated nerve ﬁbers. Physica D: Nonlinear Phenomena,
148(1–2), 159–174.
Blot, A., & Barbour, B.

(2014). Ultra-rapid axon-axon ephaptic
inhibition of cerebellar Purkinje cells by the pinceau. Nature
Neuroscience, 17, 289–295.

Bokil, H., Laaris, N., Blinder, K., Ennis, M., & Keller, UN.

(2001).
Ephaptic interactions in the mammalian olfactory system. Journal
of Neuroscience, 21(20), RC173.

Lui, B., Yang, L., Wilke, C., & Yuan, H. (2011). Electrophysiological
imaging of brain activity and connectivity—Challenges and op-
portunities. IEEE Transaction on Biomedical Engineering, 58(7),
1918–1931.

Hodgkin, UN. L., & Huxley, UN. F.

(1952). A quantitative descrip-
tion of membrane current and its application to conduction and
excitation in nerve. Journal of Physiology, 117(4), 500–544.

Holt, G. R., & Koch, C.

(1999). Electrical interactions via the ex-
tracellular potential near cell bodies. Journal of Computational
Neuroscience, 6(2), 169–184.

Katz, B., & Schmitt, O. H. (1940). Electric interaction between two
adjacent nerve ﬁbres. Journal of Physiology, 97(4), 471–488.

Lamantia, UN. S., & Rakic, P.

(1990). Cytological and quantitative
characteristics of four cerebral commissures in the rhesus mon-
key. Journal of Comparative Neurology, 291(4), 520–537.

Marrazzi, UN. S., & Lorente, R. (1944). Interaction of neighboring ﬁbres
in myelinated nerve. Journal of Neurophysiology, 7(2), 83–101.
Moradmand, K., & Goldﬁnger, M. (1995). Computation of long dis-
tance propagation of impulses elicited by Poisson-process stim-
ulation. Journal Neurophysiology, 74(6), 2415–2426.

Nagumo, J., Arimoto, S., & Yoshizawa, S. (1962). An active pulse
transmission line simulating nerve axon. Proceedings of the IRE,
50(10), 2061–2070.

Nielsen, V. K. (1984). Pathophysiology of hemifacial spasm: IO. Ephaptic
transmission and ectopic excitation. Neurology, 34(4), 418–426.
Plonsey, R. (1977). Action potential sources and their volume con-

ductor ﬁelds. Proceedings of the IEEE, 65(5), 601–611.

Rall, W.

(1962). Electrophysiology of a nendritic neuron model.

Breakspear, M. (2017). Dynamic models of large-scale brain activity.

Biophysical Journal, 2(2), 145–167.

Nature Neuroscience, 20, 340–352.

Clark, J., & Plonsey, R. (1968). The extracellular potential ﬁeld of
the single active nerve ﬁber in a volume conductor. Biophysical
Journal, 8(7), 842–864.

Clark, J. W., & Plonsey, R. (1970). A mathematical study of nerve

ﬁber interaction. Biophysical Journal, 10(10), 937–957.

Debanne, D. (2004). Information processing in the axon. Nature

Reviews Neuroscience, 5, 304–316.

FitzHugh, R. (1961). Impulses and physiological states in theoretical
models of nerve membrane. Biophysical Journal, 1(6), 445–466.
Fröhlich, F., & McCormick, D. UN. (2010). Endogenous electric ﬁelds
may guide neocortical network activity. Neuron, 67(1), 129–143.
(2016). Neuronal coupling by en-
dogenous electric ﬁelds: Cable theory and applications to coin-
cidence detector neurons in the auditory brain stem. Journal of
Neurophysiology, 115(4), 2033–2051.

Goldwyn, J. H., & Rinzel, J.

Grindrod, P., & Sleeman, B. D.

(1984). Qualitative analysis of
reaction-diffusion systems modelling coupled unmyelinated
nerve axons. Mathematical Medicine and Biology, 1(3), 289–307.

Ramón, F., & Moore, J. W. (1978). Ephaptic transmission in squid
giant axons. American Journal Physiology, 234(5), C162–C169.
Rasminsky, M. (1980). Ephaptic transmission between single nerve
fibres in the spinal roots of dystrophic mice. Journal of Physiology
350, 151–169.

Reutskiy, S., Rossoni, E., & Tirozzi, B. (2003). Conduction in bun-
dles of demyelinated nerve ﬁbers: Computer simulation. Biolog-
ical Cybernetics, 89(6), 439–448.

Rosenblueth, UN.

(1941). The stimulation of myelinated axons by
nerve impulses in adjacent myelinated axons. American Journal
of Physiology, 132(1), 119–128.

Sanz-Leon, P., Knock, S. A., Spiegler, A., & Jirsa, V. K. (2015). Math-
ematical framework for large-scale brain network modeling in
The Virtual Brain. NeuroImage, 111, 385–430.

Scott, UN. (2002). Neuroscience: A mathematical primer. New York,

NY: Springer US.

Stacey, R. G., Hilbert, L., & Quail, T. (2015). Computational study
of synchrony in ﬁelds and microclusters of ephaptically coupled
neurons. Journal of Neurophysiology, 113(9), 3229–3241.

Network Neuroscience

609

D
o
w
N
o
UN
D
e
D

F
R
o
M
H

T
T

:
/
/

D
io
R
e
C
T
.

io
T
.

e
D
tu
N
e
N
UN
R
T
io
C
e
–
P
D

F
/

4
3
5
9
5
1
8
6
7
3
2
8
N
e
N
_
UN
_
0
0
1
3
4
P
D

B
sì
G
tu
e
S
T

o
N
0
7
S
e
P
e
M
B
e
R
2
0
2
3

A mathematical model of ephaptic interactions in neuronal ﬁber pathways

Taylor, C., & Dudek, F. (1982). Synchronous neural afterdischarges
in rat hippocampal slices without active chemical synapses. Sci-
ence, 218(4574), 810–812.

Usmani, R. UN. (1987). Applied linear algebra. New York, NY: Marcel

Dekker.

Waxman, S. G., & Swadlow, H. UN. (1977). The conduction proper-
ties of axons in central white matter. Progress in Neurobiology,
8, 297–324.

Wedeen, V. J., Rosene, D. L., Wang, R., Dai, G., Mortazavi, F.,
Hagmann, P., . . . Tseng, W. Y. IO. (2012). The geometric structure
of the brain ﬁber pathways. Scienza, 335(6076), 1628–1634.