In vivo ephaptic coupling allows memory network formation

Abstract

It is increasingly clear that memories are distributed across multiple brain areas. Such "engram complexes" are important features of memory formation and consolidation. Here, we test the hypothesis that engram complexes are formed in part by bioelectric fields that sculpt and guide the neural activity and tie together the areas that participate in engram complexes. Like the conductor of an orchestra, the fields influence each musician or neuron and orchestrate the output, the symphony. Our results use the theory of synergetics, machine learning, and data from a spatial delayed saccade task and provide evidence for in vivo ephaptic coupling in memory representations.

Full text

Received: May 18, 2023. Revised: June 20, 2023. Accepted: June 23, 2023
© The Author(s) 2023. Published by Oxford University Press.
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Cerebral Cortex, 2023, 1–19
https://doi.org/10.1093/cercor/bhad251
Original Article
In vivo ephaptic coupling allows memory network
formation
Dimitris A. Pinotsis 1, 2, *,Earl K. Miller2
1Department of Psychology, Centre for Mathematical Neuroscience and Psychology, University of London, London EC1V 0HB, United Kingdom,
2The Picower Institute for Learning & Memory, Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139,
United States
*Corresponding author: Department of Psychology, Centre for Mathematical Neuroscience and Psychology, University of London, London EC1V 0HB, United
Kingdom. Email: pinotsis@mit.edu
It is increasingly clear that memories are distributed across multiple brain areas. Such “engram complexes” are important features
of memory formation and consolidation. Here, we test the hypothesis that engram complexes are formed in part by bioelectric fields
that sculpt and guide the neural activity and tie together the areas that participate in engram complexes. Like the conductor of an
orchestra, the fields influence each musician or neuron and orchestrate the output, the symphony. Our results use the theory of
synergetics, machine learning, and data from a spatial delayed saccade task and provide evidence for in vivo ephaptic coupling in
memory representations.
Key words:memory engrams; neural ensembles; working memory; synergetics; predictive coding; auto-encoders; effective connectivity.
Introduction
In recent decades there has been a paradigm shift in neuroscience.
In the past, we focused on properties of individual neurons (James
1890;Queenan et al. 2017). There is now a growing realization
that information storage and processing depends on spatially
distributed, dynamic groupings of neurons (Fujisawa et al. 2008;
Buschman et al. 2011;Yuste 2015), known as neural ensembles
(Buschman et al. 2012;Tayler et al. 2013;Pfau et al. 2013;Pinotsis
et al. 2017;Pinotsis and Miller 2017) or engram cells (Thompson
1976;Josselyn et al. 2015). Techniques like protein induction (Gor-
don et al. 1980), immediate early gene (IEG) expression (Guzowski
et al. 2005), and optogenetics (Fenno et al. 2011) allow for iden-
tification of ensemble neurons participating in memory storage
and recall (Ryan et al. 2015;Tonegawa et al. 2015b). Further,
recent experiments have found simultaneous neural ensembles
maintaining the same memory in many brain areas, something
known as engram complex (Poo et al. 2016;Roy et al. 2019). In
Roy et al. (2019), a total of 247 brain areas were mapped using
the protein cFos and IEG) Among them, 117 areas were found to
be significantly reactivated when a fear memory was recalled.
Thus, memory was not stored in a single brain area but was
dispersed in multiple areas and neural ensembles. Earlier theories
like memory consolidation (Squire and Alvarez 1995) and multiple
traces (Nadel and Moscovitch 1997) have also found that mem-
ories are stored in multiple areas forming engram complexes.
These are connected via engram pathways formed by mono- or
poly-synaptic connections (Tonegawa et al. 2015a).
The challenge, then, is in understanding how the brain forms
engram complexes. Each brain area is connected to many oth-
ers. Anatomical connectivity alone cannot be the whole story.
Hypotheses that could explain this include that engram com-
plexes are dynamically formed by emergent properties of neu-
rons like synchronized rhythms (Harris et al. 2003;Miller et al.
2014,2018;Lundqvist et al. 2018), possibly resulting from internal
coordination of spike timing (Singer 1999;Koch 2004), that allow
neuronal communication (Fries 2015;Lakatos et al. 2019;Reinhart
and Nguyen 2019), feature integration and perceptual segmen-
tation (Engel and Singer 2001;Moore and Obhi 2012). Here, we
report tests of the hypothesis that the electric fields generated
by neurons play a crucial role. We suggest that ephaptic coupling
(Anastassiou et al. 2011;Ruff ini et al. 2020) ties together the areas
that participate in engram complexes. In other words, we test
the hypothesis that memory networks include electric fields that
carry information back to individual neurons.
Direct evidence of ephaptic coupling of spiking has been found
in brain slices (Jefferys et al. 2012;Anastassiou and Koch 2015;
Chiang et al. 2019). In vitro ephaptic coupling has been found
in LFPs. Application of external electric fields resulted in mem-
brane potentials oscillating at the same frequency as the drive
(Anastassiou et al. 2011). Support for its role in forming engram
complexes comes from studies showing that neurons participat-
ing in an engram complex showed similar functional connectivity
during optogenetic activation and memory recall (Kitamura et al.
2017;Roy et al. 2019). We found that the electric fields in the
primate prefrontal cortex carried information about the contents
of working memory (Pinotsis and Miller 2022). Using data from a
delayed saccade task (Jia et al. 2017;Pinotsis et al. 2017), we built
two models: one for neural activity (Pinotsis et al. 2017a; Pinotsis
and Miller 2017) and another for the emergent electric field.
This revealed electric field patterns that varied with contents of
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2|Cerebral Cortex, 2023
working memory. Further, we found that the electric fields were
robust and stable, whereas neural activity underlying memory
showed representational drift. This latter observation suggested
the hypothesis that electric fields could act as “guard rails” that
help stabilize and funnel the high dimensional variable neural
activity along stable lower-dimensional routes.
Here we test the hypothesis that electric fields sculpt and guide
the neural activity forming engram complexes. We used a theory
of complex systems known as synergetics (Haken 1987,2012). We
also extended the single area analysis of (Pinotsis and Miller 2022)
and focused on data from two areas known to form an engram
complex, frontal eye fields (FEF) and supplementary eye fields
(SEF). FEF and SEF areas are anatomically connected (Purcell et al.
2012) and are thought to control eye movements (Stuphorn et al.
2010). Synergetics describes how complex systems (e.g. molecules,
fluids, brain etc.) self-organize. In the case of human behavior,
synergetics describe how the collective dynamics of muscles and
body parts (e.g. fingers) give rise to behavior like rhythmic hand
movement (Haken et al. 1985). We applied synergetics to under-
stand the emergence of memory representations. We performed
mathematical, i.e. pen and paper, computations and showed that
the theory predicts that electric fields guide ensemble activity. If
ephaptic coupling occurs in a brain area and exchanges memory
information with other brain areas, then ephaptic coupling will
occur in those areas too. We then confirmed our results using
Bayesian Model Comparison (BMC; Kass and Raftery 1995;Friston
and Penny 2011), Granger Causality (GC; Barnett and Seth 2014),
and Representation Similarity Analysis (RSA; Kriegeskorte et al.
2008). This suggested that the electric field enslaves neurons,
not the other way around. Applying the slaving principle (Haken
2012), we found that the electric field controls neural activity and
oscillations through ephaptic coupling (Fröhlich and McCormick
2010;Anastassiou and Koch 2015) and that this was the case
across all recording sites that participated in the engram complex.
Methods
Mathematical notation
∇h(p)≡∂h(p)
∂xand Δh(p)≡∂2h(p)
∂x2(first and second derivatives
evaluated at point p), for an arbitrary function h.h(j)≡∂jh(x,t)
∂xj
denotes the spatial derivative of order j. The subscript “0” denotes
boundary values, e.g. ΔVe
0is the value of the second derivative of
the extracellular potential Veon the exterior of the membrane.
A random process ˜
Vmfrom which the transmembrane potential
Vmis sampled is denoted by tilde with samples ˜
Vml, indexed by
l. Hat denotes the Fourier Transform (FT) of a function h,i.e.
ˆ
h(k)=FT(h)=∞
−∞ h(ρ)eikρdρ.
Task and experimental setup
We reanalyzed data from (Jia et al. 2017). The same data were
used in our earlier papers (Pinotsis et al. 2017; Pinotsis and Miller
2022b). Two adult male Macaca monkeys were trained to perform
an oculomotor spatial delayed response task. This task required
the monkeys to maintain the memory of a randomly chosen
visual target (at angles of 0◦,60
◦, 120◦, 180◦, 240◦, and 300◦, 12.5◦
eccentricity) over a brief (750 ms) delay period and then saccade
to the remembered location. If a saccade was made to the cued
angle, the target was presented with a green highlight and a
water reward was delivered. If not, the target was presented with
a red highlight and reward was withheld. Thirty-two-electrode
chronic arrays were implanted unilaterally in FEF and SEF in each
monkey. Each array consisted of a 2 ×2 mm square grid, where the
spacing between electrodes was 400 um. The implant channels
were determined prior to surgery using structural magnetic res-
onance imaging and anatomical atlases. From each electrode, we
acquired local field potentials (LFPs; extracted with a fourth-order
Butterworth low-pass filter with a cut-off frequency of 500 Hz, and
recorded at 1 kHz) using a multichannel data acquisition system
(Cerebus, Blackrock Microsystems). We analyzed LFPs during the
delay period when monkeys held the cued angles in memory.
A neural field model of ephaptic coupling
This section summarizes a theoretical model for the description
of neural ensemble activity developed earlier (Pinotsis et al. 2017;
Pinotsis and Miller 2017,2022). We modeled the activity of neural
ensembles. These are groups of neurons that maintain working
memory representations. Some results about their activity sum-
marized below involve lengthy derivations not repeated here. The
interested reader might consult earlier papers that are referenced
and the Supplementary Material.
In earlier work, we used the neural field theory (Jirsa and
Haken 1996;Coombes 2005;Deco et al. 2008;Robinson et al.
2016), (cf. Equation 4in Pinotsis et al. 2017) to describe the
evolution of the transmembrane potential or depolarization, Vm,
in neural ensembles. Currents flow along the neurons’ axons
and dendrites. Chemical energy is converted to electrical. Action
and synaptic potentials are summed up to produce an emerging
“electric potential” (EP) Vein extracellular space. The difference
of intracellular Viand extracellular Vepotentials on either side
of the membrane, Vm=Ve
0−Vi
0is the transmembrane potential
(recall that the subscript “0” denotes boundary values). The time
evolution of the transmembrane potential Vmcan be described by
a neural field model (Atay and Hutt 2004;Pinotsis et al. 2012;Bojak
et al. 2013). Figure 1(A) includes a schematic of a chronic array
implanted in a cortical area (for simplicity, 10 electrodes shown as
dots in the blue square). Each electrode is thought to be sampling
from a neural population in its proximity and we assumed that
the ensemble occupies a patch (cortical manifold) denoted by
. Activity is sampled at the locations of the electrodes. It is
thought to be generated by a neural population in the vicinity of
the electrode. To construct the neural field model, we numbered
the electrodes in a monotonic fashion (cf. the numbers in Fig. 1A).
For mathematical convenience, we also assumed that can be
replaced by a line, i.e. electrodes are all next to each other (cf.
red line at the bottom of Fig. 1A). This assumption was tested in
(Pinotsis et al. 2017)and(Pinotsisand Miller 2022). There, we found
that the model explained more than 40% of the data variance.
A second test of this assumption is discussed after Equation (3)
below. The colored curves connecting electrodes (dots on the red
line at the bottom of Fig. 1A) are schematics of Gaussian functions
that describe connectivity between electrodes and populations,
see (Pinotsis et al. 2017) for details.
Our neural field model describes transient f luctuations around
baseline, similar to spontaneous activity in large scale resting
state networks (Deco et al. 2010;Pinotsis et al. 2013;Drysdale
et al. 2017). It predicts average firing rate or depolarization, similar
to activation functions in deep neural networks (LeCun et al.
2015;Pinotsis et al. 2019). Mathematically, the neural field model
suggests that the time evolution of depolarization Vmis given by
the following equation (see also Equation 4in Pinotsis et al. (2017)):
˙
Vm=−τ−1
XVm+K∗f◦Vm+U
K∗f◦Vm=Kz,z,t,t·f◦Vmz,tdzdt(1)
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Dimitris A. Pinotsis and Earl K. Miller |3
Fig. 1. A) Neural field model and connections. Neural fields provided a quantitative way to describe each ensemble’s patterns of activity across
simultaneously recorded sites. The same model can describe different ensembles. Each electrode occupies a position on a cortical manifold (line) 
parameterized by the variable xand is connected to all other electrodes with connections whose strength follows a Gaussian profile (colored solid and
dashed lines), see (Pinotsis et al. 2017) for more details. B) Extracellular space around each neuron within the ensemble (blue cylindrical fibers). C)
Bidomain model for the electric field generated by a cylindrical fiber in a conductor. The extracellular and intracellular space are depicted by blue and
gray cylindrical fibers (see Methods for the meaning of various symbols). D) Simplified bidomain model where the measurement point is located at a
vertical distance much larger than the radius of intracellular space.
Equation (1) suggests that Vmchanges as a result of three
terms: a simple decay, recurrent inputs from other parts of the
ensemble and some exogenous, stochastic input U. We called this
neural field “deep” to distinguish this model (with learned con-
nectivity parameters) from common neural field models where
connectivity weights are chosen ad hoc. The integral appearing in
Equation (1) is defined over the cortical patch, i.e. z∈Δand t>0.
It describes how the diffusion of local recurrent input changes Vm.
Here, zparameterizes the location on a cortical patch occupied
by the ensemble, Xis an index denoting excitatory or inhibitory
populations, Kis the connectivity or weight matrix that describes
how the signal is amplified or attenuated when it propagates
between electrodes (cf. colored curves in Fig. 1A), Uis endogenous
neural input and f(h)=1
1+exp(δ(η−h)) is called transfer function.
Also, τXis the time constant of postsynaptic filtering, δis synaptic
gain, and ηis the postsynaptic potential at which the half of the
maximum firing rate is achieved, see e.g. (Pinotsis et al. 2012)for
more details.
In Pinotsis et al. (2017), we assumed that the transmembrane
potential Vm
Xis sampled from a random process ˜
Vmwith samples
˜
Vml
X,l=1,...N. We then considered a new variable Y=Vm
X−
N−1l=1

N
˜
Vml
Xand showed that Equations (1) can be reformulated as
a Gaussian Linear Model (GLM):
Y=
j
Hjwj+ε
wT
j(z)=∇h(0)
j!τX−1Δ
Kz,z(z−z)jdz
Hj=∂jVm(z,t)
∂zj(2)
where ε∼m,ss2Iand ssis the inverse precision. Note that m
is the sample mean m=N−1l=1

N
˜
Vml
X. For a detailed derivation of
Equation (2) from Equation (1) and its relation to similar models
like Wilson and Cowan (1973),seePinotsis and Miller (2022)
and Supplementary Material. In Equation (2), the functions w
are called the connectivity components and Hare the principal
axes. The connectivity components w(second line in Equation 2)
provide the connectivity matrix K(cf. Equation 1and Fig. 1A). They
describe how signal recorded from a certain electrode contributes
to LFP data (across all trials). They are of dimensionality number
of electrodes “by” the number of trials. The principal axes (last
line in Equation (2)) are matrices of dimensionality number of
time samples “by” the number of trials.They describe the average
instantaneous contribution to the LFP data across all electrodes.
Please see Pinotsis et al. (2017),Pinotsis and Miller (2022) as well
as the Supplementary Material for more details about and the
connectivity components and the principal axes.
To find the connectivity components w, we used a Restricted
Maximum-Likelihood (ReML) algorithm (Harville 1977). This opti-
mizes a cost function known as the Free Energy, F,
F=−1
2(Y−Hw)Tr2
s(Y−Hw)+ln |s2
s|+ln |s2sΠ−1|
+ZTZ+const
Π=s2sI+HTH
Z=Π−1HTY(3)
The connectivity wwas obtained by training the neural
field model given by Equation (1) using the cost function given
by Equation (3) maximizes the mutual information between
the remembered cue and the ensemble activity. This can be
thought to describe synaptic efficacy in a neural ensemble
that represents a certain stimulus. In Pinotsis and Miller (2022),
we obtained the connectivity matrices and compared them to
the connectivity obtained using two independent methods: k-
means clustering (Humphries 2011) and high dimensional SVD
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4|Cerebral Cortex, 2023
(Carroll and Chang 1970;Williams et al. 2018). This served as
a validation of the neural field model given by Equation (1). It
also provided a second test of the earlier assumption where we
replaced the cortical patch by a line (red line at the bottom of
Fig. 1A). We found that the connectivity obtained after training
the neural field model with the cost function (3)isthesameas
the connectivity found using pairwise correlations (Humphries
2011)andSVD(Williams et al. 2018).
To sum up, in previous work we showed that neural fields given
by Equations (1) can be rewritten like a GLM given by Equation
(2). We also showed how neural fields can be trained using the
Free Energy given by Equation (3) to obtain the connectivity K.In
Pinotsis and Miller (2022), we also showed that Wilson and Cowan
network models (Wilson and Cowan 1973) can also be written in
the form of a GLM and trained using the cost function (3). Here,
we will use neural field models given by Equation (1).
Below, we will consider an extension of the model (1) that will
include ephaptic coupling (interactions between emerging electric
fields produced by neural ensembles and the underlying neural
activity). Later, we will fit this extended as well as the original
neural field model to LFP data and assess which of the two models
fits LFPs better. This will test evidence for ephaptic coupling. We
first discuss the ephaptic extension of the neural field model
below.
Above we presented a model of neural activity (cf. Equations 1)
describing current f low within an ensemble. This current gener-
ates an electric field in extracellular space, Ee. This can directly
influence individual neurons, a phenomenon known as “ephaptic
coupling” (Fröhlich and McCormick 2010;Anastassiou et al. 2011;
Ruffini et al. 2020;Rebollo et al. 2021;Schmidt et al. 2021).
Ephaptic coupling describes interactions between the brain’s elec-
tric fields and neural activity, that is, interactions between Ee
and Vm.(Danner et al. 2011)and(Goldwyn et al. 2017)showed
that ephaptic effects result in perturbations (small increases) of
transmembrane potential by adding the value of the extracellular
potential on the membraneVe
0. They described these increases by
replacing Vm≈Vm+Ve
0in the term capturing local recurrent input
as a result of diffusion. In other words, they added an ephaptic
current to the diffusion current that changes the transmembrane
potential. We did the same here. We replaced Vm≈Vm+Ve
0in the
integral in Equation (1) that describes the diffusion of recurrent
input in the ensemble and obtained
˙
Vm=−τ−1
XVm+K∗f◦Vm+Ve
0+U(4)
We called this the “ephaptic model.” Compared with Equation
(1), Equation (4) suggests that the rate of change of depolarization
comprises the same three terms as before and additionally, per-
turbations due to extracellular potential Ve
0. The ephaptic model
is used twice below: first, in “Methods,”to derive the mathematical
expression of ephaptic coupling and then, in “Results,” to find
evidence of ephaptic coupling using BMC.
A model of the ensemble electric field
We saw above that current flow within the neural ensemble
generates an electric field in extracellular space Ee=−∇Ve,where
Veis the corresponding potential. In (Pinotsis and Miller 2022), we
introduced a model of this electric field based on the “bidomain
model” (Mc Laughlin et al. 2010;Goldwyn et al. 2017). Below, we
summarize the main points of this model. For more details, the
interested reader is invited to consult (Mc Laughlin et al. 2010;
Goldwyn et al. 2017;Pinotsis and Miller 2022).
We model the electric field in extracellular space very close
to the neural ensemble that generated it. The bidomain model
assumes that dendrites of cortical pyramidal cells comprising
neural ensembles extend parallelly. Although they have a compli-
cated geometry, this symmetry allows one to replace the branched
dendrites trees by a cylindrical fiber (Rall 1998). This is the same
symmetry as that of the current dipole approximation to cortical
sources widely used in human electrophysiology (Hämäläinen
et al. 1993;Nunez and Srinivasan 2006;Lindén et al. 2010).
In this model, pyramidal neurons are aligned to produce an
EF parallel to apical dendrites and receive synchronous input.
Current flowing in neurons gives rise to dipole sources (Buzsáki
et al. 2012;Pesaran et al. 2018). The extracellular space of each
pyramidal neuron is described by a cylindrical fiber (small blue
cylinders in Fig. 1B). Using the principle of superposition from
electromagnetism, extracellular spaces can be combined into a
unified extracellular space of the neural ensemble. Thus,the indi-
vidual cylindrical fibers of Fig. 1(B) (for each neuron) are replaced
by the larger fiber surrounding the ensemble (light blue cylinder
in Fig. 1C). The boundary between extracellular and intracellular
space has the same symmetry and is denoted by a gray cylinder
in Fig. 1(C).
The bidomain model assumes spatial homogeneity and tempo-
ral synchrony similarly to the well-known dipole approximation.
EF model estimates are a bound on realistic values of EF: actual
EFs will be smaller when these assumptions fail. Note that this
does not change qualitive results,like ephaptic coupling discussed
below as the extracellular and intracellular spaces can be split
into smaller parts (cylindrical fibers) where symmetry and syn-
chrony still apply.
This electric field Eeis the result of the discontinuity in the
electric potential Ve
0−Vi
0that gives rise to electric dipole sources
and transmembrane currents 1/riΔVm(Ve
0and Vi
0are the values
of the extracellular and intracellular EPs on the two sides of the
membrane). Intuitively, Eeis the potential difference over unit
distance. Alternatively, Eeexpresses the force to which an ion is
subjected to, while in extracellular space, divided by its charge
(Jackson 1999).
Because of symmetry, the extracellular field and potential
depend on two spatial variables x,y, not 3. The variable xparam-
eterizes the location on the axis of the cylinder in Fig. 1(C) and
(D) and ya direction orthogonal to this axis. According to the
bidomain model, the extracellular potential Veat a point Px,y
in the extracellular space is given in terms of the Fourier Trans-
form ˆ
Vmof the transmembrane potential Vmby the following
expression; see Equation (17) in Pinotsis and Miller (2022b) and
Supplementary Material for more details (LFP electrodes measure
potentials Ve. Thus, we can assume that the location of the LFP
electrode denoted by a star in Fig. 1B coincides with the point
Px,ywhere the electric field is evaluated.):
Vex,y=−4πσ e/σ iFT−1ˆ
Vm(k)W(k)
W(k)=I1|k|aK0|k|y
I0|k|aK1|k|a+σi/σ eI1|k|aK0|k|a(5)
Note that because of cylindrical symmetry the functions
appearing in the second line of Equation (4) do not depend on x.
They just depend on yand the ones appearing in the denominator
are evaluated for yequal to the radius of the gray cylinder, y=a
(the cylinder separating intracellular and extracellular space,like
a membrane). Here, σl,l=e,iare the extra-and intra-cellular
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Dimitris A. Pinotsis and Earl K. Miller |5
space conductivities and I0(y),I1(y),K0(y),K1(y)are modified
Bessel functions of the first and second kind (Abramowitz et al.
1988).
Ephaptic coupling, synergetics, and the stability of the
electric field
In the next two sections of the “Methods”, we include some math-
ematical arguments that motivate hypotheses tested in “Results”.
These involve analytical, ie. pen and paper calculations. Above,
we summarized a model of the electric field generated by neural
ensembles. In Pinotsis and Miller (2022b), we used this model to
compute the EF corresponding to neural ensembles maintaining
different memory representations. We found that EFs were more
stable than neural activity and contained relatively more informa-
tion. We suggested that this stability allows the brain to control
the latent variables that give rise to the same memory. In other
words,we hypothesized that EFs can sculpt and herd neural activ-
ity and can act as “guard rails” that funnel the higher dimensional
variable neural activity along stable lower-dimensional routes.
Below we provide further mathematical arguments in support
of the above hypothesis: that bioelectric fields guide neural activ-
ity. In the “Results” section, we test this hypothesis, using data
from a spatial delayed saccade task.
We were interested in interactions between variables expressed
at different spatial and temporal scales: bioelectric fields and
neural activity. Thus, we used a theory that describes interactions
underlying spontaneous pattern formation in biological and phys-
ical systems known as “synergetics” (Haken 1987;Jirsa and Haken
1996). Synergetics studies how individual parts—in our case neu-
rons—produce structures, here, memory representations. It sug-
gests that a biological system, like a neural ensemble, is con-
strained by so-called “control parameters” that impose limita-
tions. When control parameters change, the structures change.
A simple example of a control parameter is temperature. When it
changes, the state of water molecules can change from solid, to
fluid, to air. In the synergetics language, the individual elements
of the system, e.g. molecules, are called enslaved parts. This
is because they are controlled by temperature. Besides control
parameters and enslaved parts, synergetics also considers order
parameters, that is, low-dimensional descriptions of collective
dynamics, like the average transmembrane potential Vmthat we
studied here or other latent variables (Yu et al. 2008;Gallego et al.
2020) like effective connectivity components (Pinotsis et al. 2017).
A crucial distinction between control and order parameters is how
fast they evolve. When there is a perturbation, like new input
to a brain area, the order parameters and enslaved parts evolve
fast and the control parameters slowly. Control parameters are
very stable compared with order parameters. To put it differently,
synergetics suggests a temporal hierarchy comprising, slow con-
trol parameters, like temperature or energy (Ditzinger and Haken
1989), faster order parameters, and very fast enslaved parts (e.g.
oscillations/spiking (Miller et al. 2018)).
Below, we will use the theory of synergetics to provide a math-
ematical formulation of ephaptic coupling, that is, the interac-
tions between the ensemble electric field, Ee,andtheaverage
transmembrane potential Vm. We will present some theoretical
arguments that motivate the hypothesis that a slow EF Eeacts as
a control parameter, which enslaves faster neural activity Vm.In
“Results,” we will test this hypothesis and ask whether ephaptic
coupling can be detected in in vivo neural data.
To describe extracellular field—transmembrane potential, Ee—
Vm, interactions, our starting point is equations that express
one quantity in terms of the other, that is, Eein terms of Vm
and vice versa. These are Equations (4)and(5): the evolution
of transmembrane potential Vmin terms of the extracellular EP
Veis given by the ephaptic model (4). Also, Vein terms of trans-
membrane potential Vmis given by the bidomain model (5). To
perform pen and paper calculations, we need algebraic equations
(i.e. equations without the inverse Fourier transform FT−1). Thus,
in Supplementary Material,weshowhowweonecanrewrite
Equation (5) as a differential algebraic equation; see Equation (6)
below. For simplicity, we assume that the LFP electrode is at a
large distance compared with the size of the neural ensemble: the
radius aof the fiber separating the intra- and extra-cellular spaces
(gray cylinder) is very small compared with the vertical distance
yto the location of the LFP electrode, a<< y, cf. squashed gray
cylinder in Fig. 1(D).
From trial to trial, the remembered stimulus changes. Thus, the
EP and the corresponding EF also change, see (Pinotsis and Miller
2022b) for details. Assuming a fixed-point attractor (steady state),
Equation (5) can be written as (see Supplementary Material for
details)
˙
Ve=−τEP−1Ve+γ1/r−L2/12r3+L4/80r5ΔVm
Ee=−∇Ve(6)
where τ−1
EP is the rate with which Vedecays to its resting value Ve
S.
Equation (6) expresses the dynamics of the extracellular EP
Vein terms of the transmembrane potential Vm. To describe inter-
actions between these potentials and the corresponding electric
fields, we then applied the “slaving principle” from synergetics
(Haken 1987). This predicts that control parameters evolve more
slowly and constrain order parameters and enslaved parts. Exam-
ples of the general slaving principle can be found in physics and
biology (Haken 2012). Haken and colleagues have shown that
varying the temperature (control parameter) of a fluid heated
from below, various spatial patterns of f luid molecules occur. Also,
that attention can be thought of as control variable in multi-stable
perception (Ditzinger and Haken 1989;Basar et al. 2012).
During working memory delay, the “slaving principle leads to
ephaptic coupling”: it predicts that extracellular EP, Ve,enslaves
neural activity described by the transmembrane potential Vm.To
confirm this, consider the following expansion of Vmand Vein
terms of Fourier series Ve
Vm=
nξn
ψneinx. Then, substituting
these expansions into Equations (4)and(6), we obtain evolution
equations for the Fourier coefficients or modes:
˙
ξn=−τEP −1ξn+γ1/r−L2/12r3+L4/80r5n2ψn
˙
ψn=−τNA −1ψn+δ
q
ˆ
Knq ψq+ξq0+U
ˆ
Knq =Ke−i2π(nx+qy)dxdy(7)
ξnand ψnare called the Fourier coefficients or modes of the
extracellular potential and neural activity. Below, we call them
modes. ξq0are the values of the extracellular EP on the exterior
of the ensemble membrane (surface of gray cylinder in Fig. 1B).
Intuitively, a Fourier expansion implies that Vmand Veare super-
positions of planar waves einx with amplitudes given by ξnand ψn.
We have replaced Equations (4)and(5) that describe the cou-
pling between the extracellular potential and neural activity, Ve
and Vm, by Equations (7) that describe the same coupling in
terms of modes. Note that in the second equation (7), the rate
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6|Cerebral Cortex, 2023
of change of neural activity modes, ˙
ψn, depends on values of the
extracellular potential modesξq0on the exterior of the membrane
and exogenous stochastic input U.
We can now apply the slaving principle of synergetics. This
suggests that in Equations (7), one can distinguish between slow
and fast modes. Equations—as usual—provide a formalism and
motivate experimental tests, they cannot replace these tests.
In Pinotsis and Miller (2022b), we found that the electric field
was more stable than neural activity. Correlations of single trial
estimates of electric fields were higher than correlations of similar
neural activity estimates. One could thus assume that the field
modesξq0are slow and the transmembrane potential modes ψn
are fast. An independent, theoretical argument in support of this
hypothesis is a common assumption in bio-electromagnetism
about the EF being quasi-static: the tissue impedance on top
of resistance (or more generally reactance) is assumed to be
negligible and electromagnetic propagation effects can be ignored
(Nunez 1998). In other words, the electric field is assumed to
be relaxing very slowly compared with quickly relaxing neural
activity. If that hypothesis holds, the damping constant for the
extracellular potential would be much smaller than the damping
constant for neural activity τ−1
NA >> τ −1
EP (adiabatic approximation;
Haken 1987).
Using synergetics and assuming τ−1
NA >> τ −1
EP , Equations (7)
suggest that the instantaneous values of fast relaxing quantities,
like the transmembrane potential modes ψn, depend on slowly
varying quantities, like the extracellular potential coefficients ξq0
above, which slave them (Haken 1987). Electric fields enslave
neural activity. This is ephaptic coupling formulated in the lan-
guage of synergetics—as a special case of the slaving principle.
Equations (7) are then the mathematical expression of ephaptic
coupling. In Haken (1987,2012), several equations similar to (7)
are presented in the context of physics and biology and similar
coupling between fast and slow quantities is discussed.
Note that Equations (7) are “not” used for calculations in
“Results.” Had we used them in our calculations, we would
have had to prescribe the rate constants τ−1
NA,τ−1
EP a priori. This
would bias our conclusions. Equations (7) are useful because they
“motivate” a hypothesis that is tested in “Results”—independently
of Equations (7). This hypothesis is that electric field modes ξq0
are slow and neural activity modes ψnare fast.
Assuming that neural activity is enslaved by the electric field
has another implication. It suggests that instantaneous values
of neural activity are given in terms of instantaneous values of
the slower fields. During the delay period of the memory task
considered here, one can assume fixed point dynamics. In other
words, the transmembrane potential can be assumed to be in
equilibrium, thus |˙
ψn|= 0. Then, Equations (7)yieldthese
instantaneous values of neural activity determined by emerging
fields. One can express ψnin terms of ξn:
ψn=δ
τNA 
q
ˆ
Knq ψq+ξq0(8)
This equation describes how the fast modes of neural activity
are enslaved (driven) by the slow, “stable” modes of the electric
field.
To sum up, the slaving principle from synergetics predicts
that stable electric fields enslave neural activity. Mathemati-
cally, this result is expressed via Equations (8) and in neuro-
science it is called ephaptic coupling. The slaving principle distin-
guishes between stable and unstable quantities, like the modes ξn
and ψn. It suggests that the evolution of fast unstable modes is
determined by stable modes. The latter determine the instanta-
neous values of the former: here the electric field determines
neural activity, see Equation (8). This is also related to critical
slowing where some modes are strongly correlated over time (e.g.
Bassett and Bullmore 2009;Kitzbichler et al. 2009;Chialvo 2010),
see also Pinotsis and Miller 2022b).
Ephaptic coupling across engram complexes
The distinction between stable and unstable modes can be
obtained using a mathematical theory known as linear stability
analysis. Linear stability analysis of neural network models is
often used to express brain responses in terms of key anatomical
and biophysical parameters (e.g. Coombes 2005;Jirsa and Haken
1996;Pinotsis et al. 2013;Pinotsis and Friston 2010). It can also
be extended to include nonlinear terms (see Haken 1987;Basar
et al. 2012). Here, we use linear stability analysis to motivate
a hypothesis about engram storage in memory networks that
will be tested in “Results” that ephaptic coupling occurs across
engram complexes. “If ephaptic coupling occurs in a brain area
and this exchanges memory information with other brain areas
then ephaptic coupling will occur in those areas too.”Below,we
present mathematical argument in support of this hypothesis for
two areas. Generalization to an arbitrary number of areas can be
done by induction.
Consider two neural ensembles in brain areas (i) FEF and (ii)
SEF. Dynamics of ensemble activity are given by a system of neural
fields of the form of Equation (4). Similarly to Equation (4)above,
ephaptic coupling suggests Vjm depends on EP Vje
0(its boundary
value at the membrane exterior) via the following expressions:
˙
V1m=−τNA −1V1m+K1∗f◦V1m+V1e
0+U1
˙
V2m=−τNA −1V2m+K2∗f◦V2m+V2e
0+W∗f◦V1m+U2(9)
Here, Wis the feedforward connectivity matrix whose entries
are weights that scale downstream input to SEF from FEF (Gross-
berg 1967;Wilson and Cowan 1973). Note that in “Results,” we
did “not” use predicted data from model (9). This is just used
here for mathematical analysis and formulating mathematical
arguments. In the linear stability regime, we can assume that
the transmembrane potential Vim of each ensemble (identified
by the upper index j= 1,2) includes perturbations in the form of
planar waves around baseline Vio, which is an equation of the form
Vjm ∼Vjo +eβt+ikx (Pinotsis and Friston 2010;Grindrod and Pinotsis
2011). For mathematical convenience, we consider a vector of
extracellular and transmembrane potential functions for the two
areas:
Φ=V1e,V1m,V2e,V2mT=
n
⎛
⎜
⎜
⎜
⎝
ξ1
n
ψ1
n
ξ2
n
ψ2
n
⎞
⎟
⎟
⎟
⎠
einx (10)
Upper indices denote the area and lower indices the mode
order. In the previous section we saw that the slaving principle
suggests that the slow, stable field modes ξ1
nand ξ2
nwill constrain
ψ1
nand ψ2
n. The order of the expansion (10), n(how many modes are
needed to faithfully represent the dynamics), can be found using
a model fitting procedure (e.g. maximum likelihood or similar)
using real data. We will consider this elsewhere. Since we here
focus on mathematical arguments, for simplicity, we assume that
the first two modes explain most of the observed variance, that is,
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Dimitris A. Pinotsis and Earl K. Miller |7
we keep terms up to the second order in Equation (10)(n=1,2):
Φ≈ξ1
1,ψ1
1Teix +ξ1
2,ψ1
2,Te2ix +ξ2
1,ψ2
1Teix +ξ2
2,ψ2
2Te2ix (11)
Substituting the above expression in Equations (9) and using
the first of Equations (7), we obtain a system of equations,
˙
Φ=MΦ+nonlinear terms (12)
where the matrix Mcan be expressed in terms of 4x4 matrices A,
B,C,andD,M=AB
CD
defined in the Supplementary Material.
Further, the matrix Dcan be written as D=EL
GJ
in terms of
2×2 matrices E,L,G,andJalso included in the Supplementary
Material. Equation (12) is a linearized system that describes the
coupling of extracellular and transmembrane potentials in the
two brain areas in terms of connectivity matrices between them.
Mathematically, for the system to have a solution, that is, for the
modes in all areas to exist, the determinant of the matrix Mneeds
to be different than zero, det(M)= 0. Existence of a solution of a
linear system when the determinant of the coefficient matrix is
non zero is a standard result in Linear Algebra (Strang 2006). Note
that Mis called the coefficient matrix of the linearized system
given by Equation (12).
But what does existence of solution mean? Intuitively, it means
that one can find functions that satisfy these equations. In other
words, this is just a mathematical tautology that there are some
functions ψ1
j,ξ1
jand ψ2
j,ξ2
j, that is, some extracellular and trans-
membrane potentials that can describe electrical activity in FEF
and SEF. The implied assumption here is that there is also a
feedforward connectivity matrix W(recall Equation 9)sothatFEF
and SEF form an engram: there is input from one area to the other.
This is an assumption in our analysis (the areas form a memory
engram or network). To sum, Equation (12) and the condition
det(M)= 0are just a mathematical expression of the simple
fact that ensembles FEF and SEF are connected and generate
some activity and electric fields. By applying the identity det(M)=
det A−BD−1Cdet(D)(Abramowitz et al. 1988), we obtain
det(M)=det(A)det E−LJ−1Gdet(J)(13)
Thus, the condition det(M)= 0 requires that det(J)= 0 and
det(A)= 0; the determinants of matrices Jand Ashould also be
non zero. Jis defined by
J=β+τEP−14Z
δˆ
K2
22 β+τNA−1+δˆ
K2
22
Z=γ1/r−L2/12r3+L4/80r5(14)
In other words, Jis the matrix of coefficients in a linearized
system of equations describing the coupling between the sec-
ond extracellular and membrane potential modes in the second
region:
˙
ξ2
2=−τEP−1ξ2
2+γ1/r−L2/12r3+L4/80r54ψ2
2
˙
ψ2
2=−τNA −1ψ2
2+δˆ
K2
22 ψ2
2+ξ2
20(15)
Then, the condition det(J)= 0 implies that the above system
has a solution, i.e. there are some functions (modes) that describe
the extracellular and transmembrane potentials. det(J)= 0also
includes some additional piece of information. This is due to the
similarity of these equations with Equations (7). In the previous
section, we found using synergetics that Equations (7)arethe
mathematical expression of ephaptic coupling. Equations (15)
are the same as (7) when we consider the second order modes
(denoted by the lower index “2” in ξ2
2) in the second area (SEF;
denoted by the upper index “2”). Thus, Equations (15)arethe
mathematical expression of ephaptic coupling for second order
modes in the second area. In other words, if we assume that
Equations (15) hold (mathematically, a solution exists) the trans-
membrane and extracellular potential modes will be linked via
ephaptic coupling. Similarly, the condition det(A)= 0means
that the modes in the first area will also be linked via ephaptic
coupling.
We turn to Equation (13), which says that if the determinants
of the matrices Mand Jare non zero, then the determinant of
matrix Awill also be non zero (the same is true if we replace J
by A). In mathematical notation, the following statement holds: if
det(M)= 0 and det(J)= 0,thendet(A)= 0. Above we saw what
each of these three conditions means. Following these earlier
interpretations, we can put the last mathematical statement into
words: Assuming that FEF and SEF are connected and generate
activity and electric fields (det(M)= 0) and that there is ephaptic
coupling in the second area (det(J)= 0), there will be ephaptic
coupling in the first area (det(A)= 0; or the other way around,
wherewereplaceJby A). In brief, assuming that ephaptic coupling
occurs in one area, then ephaptic coupling will also occur in the
other area. By induction, we can show the same result for an
arbitrary number of areas that form a memory network or engram
complex.
Granger Causality
To test for information transfer between different spatial scales
(emerging electric fields and neural activity) and brain areas (FEF
and SEF), we used GC (Granger 1969;Geweke 1982). GC quantifies
how the history (past samples) of variable A improve prediction
of unknown samples (future samples) of a different variable B.
It is based on generalized variances or log likelihood ratios that
quantify whether a regression model including variable A fits
future samples of variable B better than the restricted regression
model based on variable B samples only (Friston et al. 2013).
Following Barnett and Seth (2014),we evaluated GC as follows:
we first used model-based VAR modeling to calculate regression
coefficients from our data, similar to a discrete stationary vector
stochastic process. First, one determines an appropriate order of
a VAR model using an information criterion or cross-validation
(Burnham and Anderson 1998). Then, a log-likelihood ratio FA→Bof
residual covariance matrices is computed. This corresponds to the
full and restricted VAR models and quantifies the “GC strength,”
that is, whether the prediction of future values of the variable B
improves significantly after including past values of A. This can
be computed using Granger’s F-test for univariate problems or
a chi-square test for a large number of variables (Granger 1969;
Geweke 1982). GC is often used for the analysis of time series
(samples are obtained using measurements at different moments
in time). Here, we used GC after considering spatial samples, that
is, we obtained measurements at different locations in the neural
ensemble and extracellular space. This is discussed further in
“Results.”
Representation similarity analysis
We used RSA (Kriegeskorte et al. 2008) to assess the similarity
of information representation across different brain areas. RSA
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8|Cerebral Cortex, 2023
uses dissimilarity matrices (DMs) to summarize how stimulus
information is represented by brain responses. Following (Pinot-
sis et al. 2019), we built DMs based on time correlations that
are thought to underlie working memory representations (Wallis
et al. 2015;Inagaki et al. 2017). Each DM entry contained the
dissimilarity between trials corresponding to different remem-
bered cued locations. Thus, DMs describe pairwise differences
in patterns of neural activity corresponding to different stimuli.
To understand whether similar information (cued location) was
encoded in different brain areas, we computed the dissimilar-
ity between brain DMs. Following Kriegeskorte et al. (2008), the
dissimilarity between DMs, known as deviation, was the corre-
lation distance (1- Spearman correlation; Spearman was used
as it does not require a linear correspondence between these
matrices contrary to Pearson correlation). Deviations between
DMs quantify matches between representation content of brain
responses (Kriegeskorte 2011). They measure the correlation dis-
tance between each DM and quantify differences of differences:
how different are the corresponding pairwise differences in neural
activity or electric fields. After calculating deviations of DM matri-
ces, one can assess significant correspondence between informa-
tion stored in different brain areas (Diedrichsen and Kriegeskorte
2017;Peterson et al. 2018).
Results
Ephaptic coupling in in vivo memory delay data
We first asked whether we could find evidence for ephaptic
coupling in our data. We examined in vivo LFPs acquired from FEF
and SEF during delay in a spatial WM task (Jia et al. 2017;Pinotsis
et al. 2017). In (Pinotsis and Miller (2022b), we analyzed the same
data from FEF only. Here we extended our analyses to the FEF-SEF
memory network (engram complex).
To assess evidence of ephaptic coupling in our data we used
computational modeling. We considered two variants of the same
model: with and without ephaptic effects (ephaptic and non
ephaptic). First, we fitted the models to LFP data and compared
their fits. Second, we used model predictions and GC to assess
evidence of ephaptic effects. This is discussed in the next section.
Below, we discuss computational models and their f its.
In an earlier work (Pinotsis et al. 2017), we obtained predic-
tions of the activity of neural ensembles maintaining different
cued locations. Transmembrane depolarization was described by
a neural field model trained as an autoencoder, which we called
a “deep” neural field. The term “deep” reflects the bottleneck
architecture of the ReML algorithm used for training. The model
was trained using the same LFP dataset as that considered in the
analyses below. We used different parts of the dataset for fitting
and training, see “Methods” for details.
Here, we obtained new predictions of the activity of neural
ensembles by extending the model of Pinotsis et al. (2017) to
include ephaptic coupling (“Methods”); see also Goldwyn et al.
(2017). In other words, our analyses below used two sets of pre-
dictions of neural activity: with and without ephaptic coupling.
Predictions without ephaptic coupling were obtained in Pinotsis
et al. (2017). Predictions of neural activity with ephaptic coupling
were obtained here following (Danner et al. 2011;Goldwyn et al.
2017). These were obtained in two steps: first, we calculated the
extracellular electric potential generated by the neural ensemble
using a model from bioelectromagnetism (bidomain model) intro-
ducedin(Pinotsis and Miller 2022), see also (Mc Laughlin et al.
2010;Goldwyn et al. 2017). Second, we added an ephaptic current
to the local recurrent input to the neural ensemble that changed
ensemble activity. The ephaptic current was an additional current
resulting from effects of the extracellular electric potential near
the ensemble.
To look for evidence of ephaptic coupling, we fitted the pre-
dictions of the ephaptic and non ephaptic models to LFP data and
evaluated goodness of their fits. Model parameters were the same
as in Pinotsis et al. (2017) and Pinotsis and Miller (2022b). These
are included in the Supplementary Table 1.WeusedBMC(Friston
et al. 2007;Friston 2008;Pinotsis et al. 2014) to f ind the model
that fit the data best (Pinotsis et al. 2018). The ReML algorithm
was used for fitting (Friston 2008). We also used previously unseen
data (data not used for training) to avoid data leakage.
We compared the evidence of the two models (how well a
model could explain the data), the ephaptic and the non ephaptic.
If the fit of the ephaptic model was better, this would provide
evidence of ephaptic coupling under the assumption that models
are plausible. The validity of the original neural field model (with-
out ephaptic coupling) has been assessed previously: variance
explained by the model was about 40%; see Supplementary Fig.
3(A) and (Pinotsis et al. 2017). In Pinotsis and Miller (2022),we
also showed that the model obtained the same neural ensemble
connectivity as that obtained using two independent methods: k-
means clustering (Humphries 2011) and high dimensional SVD
(Carroll and Chang 1970;Williams et al. 2018). The above results
support the validity of the original model (non ephaptic). We will
return to the validity of the extended model (ephaptic model),
after we discuss the results of model fits and comparison below.
To compare models and evaluate their fits, we used model
evidence. This was computed using a Free Energy approximation.
Free Energy is a cost function borrowed from autoencoders that
we used to measure goodness of fit. Inference used single trial
data and the principal axes as input to infer connectivity, simi-
lar to Dynamic Causal Modeling (DCM) and other model fitting
approaches (Freestone et al. 2014;Oesterle et al. 2020;Pinotsis
et al. 2012). Having obtained the Free Energy, one can computer
the Bayes factor (BF; Kass and Raftery 1995). BF >3 suggests that
the model with the higher Free Energy explains the data better.
BF can be thought of as a probabilistic analogue of the odds
ratio used in frequentist statistics. This corresponds to a posterior
probability of 95% for the winning model. Here, BF describes how
likely is the ephaptic model to have generated the sampled LFPs
compared with the non ephaptic model, under a fixed effects
assumption (same model for all trials).
BF results are shown in Fig. 2(A) (vertical axis). These are aver-
aged over trials for each cued location. The horizontal axis shows
the six different locations (angles) cued to hold in working mem-
ory. Black bars denote the BF after fitting FEF data, whereas
gray bars after fitting SEF data. A positive BF implies that the
non ephaptic model was more likely; a negative BF that the
ephaptic model was. The arrow at the right-hand side of Fig. 2(A)
facing upwards includes the letters NE= non ephaptic, whereas
the downwards facing arrow, the letter E = ephaptic wins. BF bars
pointing “downwards” provides evidence of ephaptic coupling.
Using model comparison, we found that in FEF, the ephaptic
model was more likely for cued locations at θ= 60, 80, 240, and
300◦(BF = −120, 70, 45, and 55, respectively; black bars in Fig. 2A).
To make sure the ephaptic model fitted single trial data better,
Fig. 2(B) shows the BF for individual trials for θ= 60, 180, 240, and
300◦, i.e. when the ephaptic model was more likely in FEF (results
for other angles are shown in Supplementary Fig. 5A). Average
BF estimates reported in Fig. 2(A) are not driven by outliers. We
confirmed that the ephaptic model was better in most trials.
BF estimates are within BF = 20–310 for θ=60
◦, BF = 10–200 for
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Dimitris A. Pinotsis and Earl K. Miller |9
Fig. 2. A) BF for different cued locations (horizontal axis). Blue bars denote the BF after fitting FEF data, while red bars after fitting SEF data. A positive
BF implies that the non ephaptic model was more likely; a negative BF that the ephaptic model was. BF bars pointing “downwards” provides evidence
of ephaptic coupling, denoted by the E inside the lower arrow. NE in the upper arrow stands for “non-ephaptic.” B) BF for individual trials and specific
cued angles when the ephaptic model wins. Different trials are shown on the horizontal axis. The corresponding cued angles are shown at the top right
corner of each plot. The ephaptic model fits the data better for most trials.
θ= 180◦and 249◦, and BF = 5–220 for θ= 300◦. In SEF (gray bars
in Fig. 2A), the ephaptic model was more likely for θ=60
◦and
300◦(BF = −10 and 20, respectively) (As expected, the complexity
of both models was very similar, see Supplementary Fig. 2C, which
shows the difference in complexity between models.All estimates
are between 0 and 0.6, which is less than 0.5% of the BF factor
shown in Fig. 2A. Note also that for θ=0
◦, the non ephaptic model
was more likely.
Individual trial data are shown in Supplementary Fig. 5(B).
Although results were robust over trials, we did not find evidence
of ephaptic coupling across all cued locations. Thus, we then
asked why there is evidence in favor of ephaptic coupling for some
cued locations and not others. Either there was no ephaptic cou-
pling in these cases or the model overfitted. The first explanation
is refuted by the results of the next section that assesses evidence
of ephaptic coupling using a different method, GC. The second
explanation is consistent with these results and also follows from
a careful consideration of model predictions—which also reveals
limitations of the non ephaptic model.
We saw above that the original model was found to predict neu-
ral activity and connectivity when tested against the LFP data and
other methods (Briefly, it explained 40% of the variance and found
the same connectivity for ensembles maintained cued locations,
see above.). The ephaptic model includes small perturbations of
transmembrane potentials due to extracellular field effects. We
thus focused on these perturbations that we call ephaptic effects
(on neural activity). To find them, we subtracted the predictions of
the non ephaptic model from the corresponding predictions of the
ephaptic model (averaged over trials). The models predict fluctu-
ations of neural activity around baseline because of endogenous
noise driving the neural ensemble in the form of transient non
Turing patterns (patterns that decay back to baseline).
Ephaptic effects are included in Supplementary Fig. 1.Supple-
mentary Fig. 1(A) (left) shows the relative percent changes due
to the ephaptic coupling for FEF. Similarly, Supplementary Fig. 1B
(right) shows the corresponding relative changes for SEF. There
are six panels in each figure, each corresponding to a different
cued location (angle). This is shown in bottom right of each panel,
e.g. the top left panel corresponds to cued location θ=0
◦.The
vertical panel axes show the relative change in principal axis
strength with respect to the original principal axis, after including
ephaptic coupling. The horizontal panel axes show time in ms.
Ephaptic effects (amplitudes of neural activity) are expressed as
relative increases in amplitude with respect to fluctuations when
ephaptic coupling is not considered (i.e. predictions of the non
ephaptic, original model).A positive relative change of α%implies
that the amplitude of neural activity is α% larger (or smaller if the
change is negative).
Comparing Fig. 2(A) and Supplementary Fig. 1, we concluded
that the ephaptic model explained the FEF data better only when
ephaptic effects were small, i.e. below 40% and cued locations at
θ=60
◦, 180◦, 240◦. Effects for the case of the remaining two cued
locations for θ=0
◦and 120◦are up to 200% (two times larger).
Small ephaptic effects suggest that potential modulations do not
alter the homeostatic stable point and the excitation to inhibition
balance is maintained (Turrigiano 2011). This also ensures the
ephaptic model is operating within its stable (linear) regime. Sim-
ilarly, the model explained the SEF data better for cued locations
at θ=0
◦,60
◦, and 300◦when effects were small, i.e. below 6%
relative change.For the remaining cued locations at θ= 120◦,180
◦,
and 240◦, ephaptic effects were up to 600% (six times larger).This
suggests that the ephaptic model overfitted large fluctuations—
which can be explained from the linearity assumption (Taylor
expansion) inherent in its derivation (see Supplementary Material
and Pinotsis et al. 2017).
Below, we did not use the ephaptic model any further. This
was only used again in “Methods” to carry out a pen and paper
i.e. analytical, derivation of Equations (7)and(12). It was used
to formulate mathematical arguments in support of hypothe-
ses tested in “Results” (see “Methods”). Below, we only used the
original, non ephaptic model and GC. GC allowed us to test for
“nonlinear” interactions between the electric f ields and neural
activity. This was a second way to assess if there is evidence of
in vivo ephaptic coupling in our LFP data (the first was model
comparison above). Crucially, GC also allowed us to obtain the
directionality of these interactions. Comparing models above,
did not directly assess directionality. We turn to GC analyses
below.
Top down information transfer from emerging
electric fields to neuronal ensembles
Above, we found that, when endogenous f luctuations were small
(fractions of fluctuations of membrane potential around base-
line), a model in which neural ensemble activity is coupled to
the electric field (ephaptic model) explained the LFP data bet-
ter than a model without ephaptic coupling. We next tested
for ephaptic coupling more generally, during large endogenous
fluctuations. To do so, we used predictions of neural activity
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10 |Cerebral Cortex, 2023
from the non ephaptic model considered earlier and GC (see
also “Methods”). GC is a data-driven method for determining the
directionality of information flow between stochastic variables
(Granger 1969). Crucially, GC provides the directionality of the
interactions between the electric field and neural activity. In other
words, GC allows us to test whether the electric field guides
neural activity or the other way around. In Pinotsis and Miller
(2022), we suggested that electric fields can act as “guard rails”
that funnel the higher dimensional variable neural activity along
stable lower-dimensional routes. Here, we tested this hypothesis
directly using GC.
Besides the non ephaptic model that gave us predictions of
neural activity (Equation 1), we also used a model of the electric
field, known as the bidomain model, see Equation (5)andrelevant
discussion in “Methods”andSupplementary Material for details.
Model parameters for both models are included in Supplementary
Table 1. This model provides predictions of the electric field gen-
erated by neural ensembles maintaining different cued locations.
This is the near field in extracellular space, in the vicinity of the
brain tissue occupied by the neural ensemble. Taken together, the
non ephaptic and the bidomain model provide two time series,one
for predictions of neural activity and another for electric fields.
We used the non ephaptic model to get neural activity because its
predictions were shown to explain a large part of data variance
and to correlate with other methods (see the previous section and
footnote 3). Also, the model does “not” a priori assume ephaptic
coupling, to avoid biasing results. As with any model, it is just an
approximation of the observed neural activity—and similarly for
the electric f ield model. Possible interactions between predictions
of the neural and electric field models suggest that such inter-
actions could occur in the brain too. Also, the use of GC allowed
us to assess nonlinear interactions not considered in the BMC
above. Note that we did not use Equations (7) for our results below
(because they include rate constants τ−1
NA,τ−1
EP and prescribing
them a priori would bias our conclusions; see the discussion
in “Methods”). We just used the models given by Equations (1)
and (5).
Having obtained two time series for neural activity and electric
fields, we assessed causal interactions between them using GC. In
its common use, GC is applied to time series data and assesses
whether knowing the past of one variable (A) helps predict the
future of another variable (B) better than just using the past of B
alone. If so, one concludes that information flows from variable A
to B. Flow is thought to occur over time, similarly to the flow of a
water molecule that f lows in a river. In neuroscience, GC is used to
describe how information flows in the brain, using sampled time
series from different areas (Barnett and Seth 2014).
One way to compute GC is by first calculating the covariance
function, that is, how strongly a time series is related to itself or
another time series. This requires psamples, that is, measure-
ments at ptime steps earlier or later (Friston et al. 2013). Implicit
in this calculation, there is an assumption of finite p, or, that the
information flows at a “finite speed” from the variable A to B.
Here, we focused on the information flow between the electric
field and neural activity (i.e. the electric field and neural activity
are the variables A and B). It is well known that interactions
involving the electric field transfer information very close to the
speed of light, which is practically “inf inite.” Thus, the assumption
of finite velocity inherent in GC analyses does not hold here.
See also the discussion in “Methods” and Equation (8). There
we presented some theoretical arguments based on the slaving
principle from complex systems theory (Haken 1987,2006). This
suggests that instantaneous values of neural activity would be
determined by electric fields, and interactions would happen at
the speed of light.
Thus, our GC analysis should be able to deal with field effects
being transmitted with practically infinite velocity. This is similar
to applications in geophysics where GC and recordings of the
earth’s gravitational field are used to, e.g. find what kind of
minerals exist deep below the surface (Marques et al. 2019). Here,
the emerging electric field contains instantaneous information
about neural activity in the same way that the gravitational field
contains instantaneous information about the masses of minerals
underneath. We used this idea from geophysics after replacing
the gravitational with the electric field and mineral masses with
neural activity.
Because of the practically infinite speed of information prop-
agation (there can be no past or future in time series data of
electric field recordings), we followed a slightly unusual GC anal-
ysis where we replaced time with space samples. We considered
snapshots of time series and computed the GC over space. We
used data from a single time point. Each snapshot corresponded
to each time point of the time series. Data included the spatial
profiles of neural activity and contemporaneous electric field
snapshots. We arbitrarily chose one edge of the cortical patch
as its beginning and the other as its end. Starting from the
beginning, we included all past locations (similar to classical
GC where past time points are used) and asked whether know-
ing the electric field helps predict the value of neural activity
in a neighboring (“future” or unknown) location, where activ-
ity had not been measured yet, better than using recordings of
neural activity alone. This is the same idea as in common GC,
where we have replaced time with space. GC measures interac-
tions between time series in both directions; thus, our analyses
answered the reverse question too: whether knowing neural activ-
ity helps predict the electric field. Our analyses are summarized
in Fig. 3.
Following Barnett and Seth (2014), we used an F-test to assess
CG strength “(Methods).” First, using LFPs from FEF, we calculated
the GC strength and