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J Comput Neurosci. Author manuscript; available in PMC 2006 December 28. Published in final edited form as: doi: 10.1007/s10827-005-0214-5. | PMCID: PMC1752229 NIHMSID: NIHMS6197 |
A Model of the Effects of Applied Electric Fields on Neuronal
Synchronization EUN-HYOUNG PARK* The Krasnow Institute for Advanced Study, George Mason University, Fairfax, Virginia 22030 ERNEST BARRETO and BRUCE J. GLUCKMAN Department of Physics and Astronomy and The Krasnow Institute for Advanced Study, George Mason University, Fairfax, Virginia 22030 STEVEN J. SCHIFF Department of Psychology and The Krasnow Institute for Advanced Study, George Mason University, Fairfax, Virginia 22030 PAUL SO  Department of Physics and Astronomy and The Krasnow Institute for Advanced Study, George Mason University, Fairfax, Virginia 22030, Email: paso/at/gmu.edu We examine the effects of applied electric fields on neuronal
synchronization. Two-compartment model neurons were synaptically coupled and
embedded within a resistive array, thus allowing the neurons to interact both
chemically and electrically. In addition, an external electric field was imposed
on the array. The effects of this field were found to be nontrivial, giving rise
to domains of synchrony and asynchrony as a function of the heterogeneity among
the neurons. A simple phase oscillator reduction was successful in qualitatively
reproducing these domains. The findings form several readily testable
experimental predictions, and the model can be extended to a larger scale in
which the effects of electric fields on seizure activity may be simulated. Keywords: synchrony, electric fields, phase response, seizure, control There has recently been a growing interest in states of sustained activity
within neuronal networks, and it has been suggested that such states may form the
dynamical underpinnings of working memory ( Wang,
2001), orientation tuning ( Hansel and
Sompolinsky, 1996), maintenance of head direction ( Sharp et al., 2001), and perhaps other neuronal phenomena
such as ocular saccade control ( Aksay et al.,
2001). States of sustained activity have also been implicated in pathological
phenomena. Epileptic seizures, for example, are pathological dynamics whose hallmark
is an abnormally prolonged and computationally dysfunctional state with sustained
activation of groups of neurons. Although it has long been believed that seizures
are highly synchronous events ( Penfield and Jasper,
1954; Kandel et al., 1991), recent
work calls this into question ( Netoff and Schiff,
2002). Interestingly, recent theoretical treatments of models exhibiting
sustained working memory activity have also suggested that such activity might be
supported by collectively asynchronous states ( Gutkin et al., 2001). The ability to interact with and modify such collective neuronal dynamical
states would be of great interest. Steady DC electrical fields applied to brain
tissue have been shown to transiently suppress seizure activity ( Gluckman et al., 1996). More recently, fields adjusted
with a simple adaptive feedback algorithm have been employed to achieve more
sustained seizure suppression ( Gluckman et al.,
2001). It remains unclear, however, how such mechanisms achieve this effect. There is at present no adequate theoretical or numerical framework available
to study how macroscopic electric fields modulate complex neuronal dynamics. It is
not even clear what sort of state modulation—increasing or decreasing
synchrony, for example—is required to achieve seizure suppression. Such
a framework would find application not only in the design of seizure suppression
technology, but also in the design of neural prosthetics where neuronal interactions
need to be carefully manipulated. As a first step towards achieving this goal, we have designed a simple model
system to study how electric fields affect synchronization in model neurons. We use
a quasi-one-dimensional array of neurons to emphasize the spatial structure of
neurons in the direction parallel to the surface of the brain. An important feature
of our model is a resistive network, which represents the conductive brain tissue in
which the neurons are embedded ( Traub, 1985).
An applied electric potential perpendicular to this array simulates electrodes
placed on the brain for the application of electric fields and for the measurement
of extracellular local differential voltages. The paper is organized as follows. In Section 2, we describe the
computational neuronal model used in our simulation and we give some mathematical
background for our subsequent phase analysis. In Section 3, we present our results,
as well as an analysis of our observed synchrony transition using a reduced phase
model. In Section 4, we summarize our results and propose some neurophysiologically
relevant predictions that might be testable in future experiments. A detailed and
complete description of our numerical model and the parameters we used can be found
in the Appendices. A report of related preliminary results has appeared previously ( Park, 2003). 2.1. Two-Compartment Neurons Embedded in a Resistive Array We based our numerical experiments on a computational model that we have
explicitly designed to include electric field interactions ( Gluckman, 1998). The presence of an electric field
induces spatial polarization in neurons ( Chan and
Nicholson, 1986; Chan et al.,
1988; Tranchina and Nicholson,
1986), and therefore, the minimal individual neuronal unit must have at
least two spatially separated compartments. We therefore chose the
two-compartment Pinsky-Rinzel (PR) model neuron, which consists of a dendrite
and a soma compartment separated by a finite conductance gc ( Pinsky and Rinzel, 1994). A
schematic representation of the PR model neuron is shown in . The dendritic compartment contains a
passive leakage current, a capacitive current, and three voltage-dependent ionic
currents. The soma compartment contains only two voltage-dependent ionic
currents, as well as passive leakage and capacitive currents. Steady currents
can also be injected independently into each compartment. For the present work, two such neurons are embedded in parallel within an
array of resistors which models the extracellular medium; see . The terminal resistors on both ends of the
array are chosen to match the impedance characteristics of an array of infinite
extent. An externally applied electric field is modeled by imposing a potential
difference Vapp between point ‘A’ and ground. The existence of
this resistive array also allows the neurons to communicate ephaptically.
Finally, the neurons are synaptically coupled to each other via both NMDA and
AMPA channels. Complete details regarding the resistive array, various currents,
rate functions, and other model parameters are presented in the Appendices. The focus of our investigation is to study the synchronization
properties of a heterogeneous network of neurons under the influence of an
applied electric field. To accomplish this goal, we modulate the applied
potential difference Vapp and we adjust the degree of heterogeneity within the network by
varying the internal conductance gc between the neurons’ somatic and dendritic compartments.
The entire system of differential equations was integrated using a 4th order
Runge-Kutta algorithm with a fixed integration time step. 2.2. The Measurement of Phase from Spike Times and the Synchrony Measure Throughout this study, we specifically define neuronal synchrony in
terms of the phase locking between the neurons’ spiking activity. In
general, uncoupled heterogeneous neurons will spike at different rates. When
these neurons are coupled together, one would expect these different spiking
frequencies to “pull” toward a common mean frequency if
the heterogeneity is not too large ( Winfree,
1980; Kuramoto, 1984). The
spike timings of the coupled neurons are expected to
“phase-lock” to each other with a possible non-zero
phase lag Ψ 0. With still stronger coupling, the
(non-identical) neurons might also spike at nearly the same time, i.e.,
Ψ 0 → 0, in addition to spiking at the
same rate. In this study, we consider two heterogeneous neurons to be
synchronized if they simply phase lock to each other. We do not impose the
stronger condition of exact synchrony with zero phase lag. The goal of this
study is to examine the relationship between the phase-locked state, the applied
electric field, and the degree of heterogeneity among the neurons. Our procedure for assigning a phase to a given neuron’s
activity is as follows. A voltage threshold for the somatic voltage Vs is chosen, and the spike times tn , where n = 1, 2, 3. . . , are defined as
the times when Vs crosses this threshold. 1 Then,
the phase corresponding to the activity of neuron i at an
arbitrary time t between its nth and
( n + 1)th spike ( tn(i) ≤ t < tn+1(i)) is defined as  . Thus, the phase increases linearly by 2π between
successive spikes (see ). A general n:m phase-locked state can be defined by the
condition | n![[var phi]](/corehtml/pmc/pmcents/x03C6.gif) 2 −
m1 −
Ψ 0| < ![[var epsilon]](/corehtml/pmc/pmcents/epsiv.gif) , where
n and m are integers ( n,
m = 1, 2, 3 . . .), 1
and 2 are the phases of the two neurons,
Ψ 0 is a constant phase lag between 0 and
2π , and is a small constant ( Rosenblum et al., 1996; Pikovsky et al., 2000). 2 For the parameter range that we examined, the 1:1 phase-locked
state was observed predominately, i.e.,
| 2( t) −
1( t) −
Ψ 0| < . In our
analysis, we introduce the relative phase Ψ( t)
= 2( t ) −
1( t) between the two neurons. From
the definition of phase given above, Ψ( t) is simply
the instantaneous phase of neuron-2,
2( t ), when neuron-1 completes its
k-th cycle at time t =
tk(1): where   is an average over the number of
spiking events ( N ), which we set to 10000 in most of our
simulations. If one assigns a two-dimensional unit vector (a phasor) to each
relative phase measurement of Ψ, then γ gives the
magnitude of the vector sum of all the phasors (see ). One can imagine that in the incoherent
case, all measurements of Ψ( tk(1)) and the corresponding phasors will be uniformly
distributed on a unit circle and γ should approach zero for large
N. On the other hand, if the neurons are phase-locked, the
distribution of the relative phase Ψ( tk(1)) will cluster around the locked value,
Ψ 0, and γ should approach its maximal
value of one (see ). 3.1. Natural Frequency Mismatch In this section, we present results on the synchronization
characteristics of our coupled neurons as the strength of the externally applied
electric field and the heterogeneity parameter Δ gc are varied. illustrates
the variation of the synchrony index γ as the applied electric field
is modulated in the case of two nearly identical neurons (solid squares) and two
very different neurons (open circles). For inhibitory (negative) electric
fields, the neurons tend to phase-lock with each other, and there is a range of
moderate inhibitory field where the spiking ceases (see gap in the figure).
Larger negative fields can depolarize the dendrites and paradoxically
“excite” the neurons back into phase-locked spiking
behavior. 3 Qualitatively, this response
to negative inhibitory fields is consistent across different levels of parameter
mismatch. However, with excitatory (positive) electric fields, we observed a more
complex network synchrony response as a function of the applied electric field
and of heterogeneity. Neurons with a small parameter mismatch first
desynchronize at moderate excitatory fields, and then resynchronize at larger
excitatory fields. 4 Neurons with a large
parameter mismatch become desynchronized as the strength of excitation becomes
stronger, and remains so. This result is consistent with the report of Golomb and Hansel (2000), who found that in
sufficiently heterogeneous ensembles, increasing coupling tends
to desynchronize the network. We show in tracings of the somatic voltage from the two neurons at
large excitatory fields ( E = 600 mV/cm) in the
synchronous and the asynchronous cases just described. In the synchronous case
(), the spiking from the two
neurons phase-lock, and this behavior is reflected in the constant
relative-phase (solid dots) plotted in the graph. In , we illustrate the asynchronous case in
which the relative phase between the two neurons monotonically drifts in time. A more complete illustration of our network’s synchronous
behavior is presented in . In this
figure, we show the γ index as a function of both the applied
electric field strength and the parameter mismatch Δ gc. We discuss several prominent features. An extensive asynchronous
region is present for large excitatory fields and large neuronal disparity, in
agreement with Golomb and Hansel (2000).
Moderate inhibitory fields tend to induce phase locking even when neuronal
disparity is large. Larger inhibitory fields suppress collective network
activity. This is consistent with in vitro seizure suppression
experiments ( Gluckman et al., 1996, 2001). Lastly, the reemergence of
phase-locked spiking activity under the largest negative fields is also seen.
Among these general characteristics, the most interesting feature of this phase
diagram is the boundary between the phase-locked and asynchronous
(phase-drifting) states under an applied excitatory field. For small neuronal
disparity there is a regime (see the wedged region indicated in ) where moderately strong excitatory fields
desynchronize the network, but an even stronger field leads to
resynchronization. The threshold for this transition at small neuronal
disparities is seen in to
increase with increasing Δ gc. We now argue that the transition to synchrony in the network including
the wedged region can be explained using a simple phase oscillator model ( Winfree, 1980; Cohen et al., 1982; Kuramoto, 1984; Ermentrout and
Kopell, 1984). Specifically, the boundary between the synchronous and
the asynchronous states is highly correlated with the degree of
natural frequency mismatch among the individual units within
the network. To illustrate this connection, we plotted the degree of natural
frequency mismatch Δω between the neurons in
isolation as a function of the applied electric field and
neuronal disparity in . For each
neuron in isolation, its intrinsic natural frequencyω is measured
with the same parameters as in the coupled network and subjected to the same
applied electric field. The natural frequency mismatch Δ
ω is then simply the difference in theω values between
the neurons. By comparing , one can see that there exists a critical value of natural frequency
mismatch Δω * such that for Δω >
Δω *, the network is in the asynchronous state, and for
Δω <
Δω *, the neurons phase lock to each other. The critical value
Δω *
was empirically determined in our numerical experiment to be approximately 0.63
rad/s. It is remarkable that this simple criterion reproduces the observed
synchrony transition boundary so well. In particular, it correctly predicts that
for small neuronal disparity (Δ gc
< 6%), increasing the (excitatory) electric
field first desynchronizes and then resynchronizes the network (see the wedged
region in ). To clarify this
point, (lower panel) shows the
decrease in Δω as a function of the increasing
(excitatory) electric field strength at Δ gc = 4%. One can understand this result by noting
that each individual neuron’s spiking rate is a monotonically
increasing concave-down function with respect to the applied electric field (see
—top panel). This
behavior is biologically reasonable since the spiking rate of a neuron will
ultimately be limited. Thus, as the spiking rate increases with increasing
excitatory fields, the corresponding incremental change between
two different neurons is smaller. In the small disparity regime, an initial
frequency mismatch might be larger than Δω * at low excitatory fields (so
that the network is asynchronous when coupled), but it might become small enough
with increasing excitatory fields such that the difference between the spiking
rates of the two neurons decreases below the critical level
Δω *. Thus, at sufficiently high excitatory field, phase
locking is observed. In our discussion so far, we have focused on the simplest 1:1 phase
locked state. Within the parameter range of our study, we have observed a few
cases where the two neurons are synchronized at higher-order n :
m phase locked states (cross-hatched squares in ). If Ω 1 and
Ω 2 are the observed frequencies of the
coupled neurons, the n : m phase locked
state is characterized by the condition:
nΩ 2 −
mΩ 1 = 0 ( Pikovsky et al., 2001). These higher order locked
states are expected to occur when the natural frequencies of
the two neurons in isolation are roughly in the same rational ratios,
i.e.,ω 1/ω 2
≈ n/m ( Ermentrout,
1981; Pikovsky et al., 2001).
gives some examples of the
values of Δ gc (10%, 14%, and 24%) at
E = −100 (mV/cm) where the natural
frequencies of the neurons give the approximate rational ratios 3/4, 2/3 and
1/2. To demonstrate these n:m phase locked states, we plotted
ΔΩ = nΩ 2
− mΩ 1 as a function of
Δ gc for four different combinations of n and
m (see ). Since
Δ gc is functionally related to Δω and is a direct
measure of the difference between the individual uncoupled neurons,
Δ gc serves as the detuning parameter for this analysis. The
n:m phase locked state in these detuning curves are indicated
as plateau regions with Δ Ω = 0. As one can
see from , the plateau regions
correspond to ranges in Δ gc where
ω 1/ω 2
≈ 1(0% ≤ Δ gc ≤ 4%), 1/2 (22% ≤
Δ gc ≤ 26%), 2/3(13% ≤
Δ gc ≤ 14.2%), and 3 /4
(9.9% ≤ Δ gc ≤ 10.3%). 53.2. Phase Response In our network, the dynamics of the individual units are determined by
the Pinsky-Rinzel equations, which include many state variables. Nevertheless,
the resulting dynamics in our chosen parameter range are predominately periodic
spikes. This periodic behavior can typically be reduced to a simple phase
equation under a suitable parameterization and the assumption of weak coupling.
Various authors ( Rand and Holmes, 1980;
Cohen et al., 1982; Kuramoto, 1984; Ermentrout and Kopell, 1984) have given detailed descriptions for
similar phase reductions in different biological systems. For the following
discussion, we assume that the reduced phase equation for the
ith uncoupled neuron can be written as where i is the phase of the ith neuron, and ω i ( E, gc(i)) is the natural frequency of the ith
neuron in isolation (i.e., without any synaptic or electrical couplings between
neurons). Note that ω i depends explicitly on the externally applied electric field
E and the internal conductance gc. With two or more neurons mutually coupled together as in our network, we
follow Refs. ( Kuramoto, 1984; Cohen et al., 1982; Ermentrout and Kopell, 1984) and further assume that
the input from the presynaptic neuron is not so strong that it significantly
alters the form of the spike. From our numerical simulations, this is a valid
assumption for almost the entire range of parameters examined. Under this
assumption, the network of coupled phase oscillators can be written as where the sum is over all neurons ( j) which are
connected to the ith neuron. In general, we expect the coupling
function fi ( E, j − i) to depend on the applied electric field E and the
relative phase of the neurons. In the case of only two mutually coupled neurons,
we can explicitly write down the following reduced dynamical equations: where Ψ = 2
− 1 is the relative phase between neuron-1
and neuron-2. To understand the dynamics of phase locking within this network, one can
consider the temporal evolution of the relative phase Ψ by
subtracting the previous two equations, where Δω ![[equivalent]](/corehtml/pmc/pmcents/equiv.gif) ω 2
− ω 1 is the mismatch in the natural
frequencies of the two individual neurons in isolation and
Δ f f1
− f2, the phase sensitivity
function, is simply the difference between the coupling functions.
For identical neurons with symmetric coupling,
f1(ψ) =
f2(− ψ) =
f (ψ) and Δ f
(ψ) will simply be twice the odd part of f
(ψ), i.e., 2 fodd(ψ)
= f (ψ) − f
(−ψ). For non-identical neurons,
Δ f (ψ) will typically contain both even
and odd parts. For a pair of neurons to phase lock to each other, the necessary
condition is  , i.e., the relative phase Ψ is constant. For a
particular choice of the parameters E and
Δ gc, the natural frequency mismatch of the neurons,
Δω, is constant and Δ f is a
function of the phase difference Ψ only. Requiring the right hand
side of Eq. (2) to be zero, the
condition for phase-locking becomes where Ψ* is the locked phase lag between the two
neurons. A schematic illustration of this phase-locked criterion is given in
. The dashed line is the constant
value of Δω and the solid line is the function
Δ f (Ψ). The crossing points (indicated
by the three small circles) are the possible phase-locked equilibrium states
predicted by this reduced phase model for a particular choice of applied
electric field E and Δ gc. The stability of these phase-locked states is given by the sign of
the local slope of Δ f (Ψ) at these
locations. For example, Ψ* b is a stable equilibrium with  . One can understand the stability argument by noticing that
for Ψ slightly less than ψ* b, Δ f (ψ)
< Δω so that  . Then, in time, Ψ will increase toward Ψ* b from the left. Similarly, for Ψ slightly larger than Ψ* b, Δ f (Ψ) >
Δω and  so that Ψ will decrease toward Ψ* b from the right. By a similar argument, one can see that the remaining
two phase-locked states at ψ* a and ψ* c are unstable equilibria. In summary, if one can obtain the phase
sensitivity function Δ f (Ψ) and the
natural frequency mismatch Δω for the ensemble at a
given choice of E and Δ gc, one can then use the reduced phase model to predict the stable phase
lags Ψ* b for the two locked neurons from a graph similar to . To get a conceptual picture of the synchrony transition for a pair of
heterogeneous neurons, we discuss a canonical example for illustrative purposes.
Consider the simple example in which f and
Δ f are given by . The coupling function f is
basically given by its canonical form of ( a0
+ a1 cos(ψ) +
b1 sin(ψ)) ( Ermentrout, 1996) and Δ f
is simply the anti-symmetric part of f. Furthermore, since the
elicitation of spikes in one neuron from another neuron cannot be instantaneous,
this coupling function f will typically be phase shifted by a
small amount δ. One should in general expect this phase shift to be
influenced by the synaptic time constants of the NMDA or the AMPA channels, the
time constant in the kinetics of the membrane, and the conduction velocity
between neurons (not modeled here). To analyze the phase-locked states of this idealized model, we need to
compare Δ f (Ψ) with
Δω. For identical neurons, the natural frequency
mismatch Δω is zero. Then according to Eq. (3), there will be two phase-locked
equilibria at Ψ* = 0(2π ) and
π (). Of these two
phase-locked states, only the solution at Ψ * = π has a
positive slope. Thus, Ψ * = π (the anti-locked state) is the
only stable equilibrium. For non-identical neurons, Δω
will be non-zero and its value will depend on the parameter mismatch
Δ gc between the two neurons and the applied electric field
E. Depending on the sign of Δω, the
relative phase Ψ *
of the stable phase-locked state will either decrease
(Δω < 0) or increase
(Δω > 0) away from the anti-locked phase. There
are two special relative phase values Ψ Lc and Ψ Rc corresponding to the left (negative) and the right (positive) peaks of
the Δ f (Ψ) function. These are,
respectively, the smallest and the largest locked relative phases possible for
the system. The critical values for this coupling threshold are achieved when
Δω = Δ f (Ψ Lc) or Δ f (Ψ Rc). These two values also define the range of allowed frequency mismatch
(Δ f (Ψ Lc) ≤ Δω ≤
Δ f (Ψ Rc)) where phase-locking between the two neurons is possible. For
Δω outside this range, a non-locked phase drifting state
between the two neurons is expected. One should note that, for non-identical
neurons, an exact phase-locked state with Ψ = 0 is not
expected to be stable in this simplified model. Biologically, this is a
reasonable conclusion since synaptic coupling cannot be instantaneous, and no
two neurons are exactly identical. The key advantage of this phase model description is that it predicts
the synchronization characteristics of the coupled network from the knowledge of
the individual neurons, namely: their natural frequencies and coupling
functions. We now apply this formalism to our model system. For a given level of
applied electric field E and internal conductance gc, the natural frequency ω( E, gc) for a particular neuron can be determined simply by taking the
inverse of the average of the spiking rate. 6 The coupling functions f1,2( E , j − i) in the reduced phase model can be rigorously calculated using the
averaging technique described in Refs. ( Ermentrout and Kopell, 1984; Hansel
et al., 1995; Ermentrout,
1996). In brief, suppose a pair of neurons is described by the following
dynamical equations: where X is a multi-dimensional vector describing the
state of the neuron, F1 and
F2 prescribe the intrinsic dynamics of the neurons
in isolation, and G1 and
G2 are the coupling functions between the two
neurons. It can be shown that when the long-term dynamics of the individual
neurons are stable limit cycles and the coupling strength is
small, then the above nonlinear equations can be reduced to a pair of phase
equations as in Eq. (1). The
coupling functions f1,2 in the phase equations can then be expressed as a
time-averaged quantity involving the original coupling functions
G1,2 as follows: where X0( t ) is the
period- P limit cycle of neuron j in
isolation and X*( t), called the adjoint solution, is
determined by the linearized dynamical equation where [ DX
F] T is the transpose of the Jacobian matrix of
F( X) with respect to the state variable
X. X*( t) is then further normalized
according to the condition X*( t)
· X′ 0 ( t)
= 1, where the prime denotes the rate of change of the vector field
along the periodic orbit X0( t).
This procedure is elegantly implemented in the publicly available software
package XPPAUT ( Ermentrout, 2002). Alternatively, one can measure the coupling function f
( E, j − i) of one neuron with respect to another by a sampling method. One can
employ a pair of unidirectionally coupled subsystems. For two mutually coupled
neurons, we decompose the system as shown in . In , neuron-1
receives synaptic input from neuron-2 only and one measures the rate of change
of neuron-1’s phase,  concurrently with 1 and
2. Then, the quantity  gives an estimate of
f1( E, 2
− 1) for a particular phase difference
Ψ = 2 −
1. In some cases, the two neurons phase lock with
each other under uni-directional coupling. In this latter situation, one can
still sample f1( E,
2 − 1) by
periodically perturbing the pair of neurons by a sufficiently large applied
electric field or by injecting current, and measuring  during the system’s transient back toward the
phase-locked state. is an example of the
coupling function calculated using Eq.
(4) for a PR neuron embedded within a resistive network with an
excitatory applied electric field. The inset is a close approximation to this
function using the sampling method described above. Both of these methods agree
qualitatively with the canonical functional form of the coupling function, and
the δ phase shift due to synaptic delay can be clearly seen in the
graph. Since the neurons are coupled both synaptically and electrically through
the resistive network, one can in principle tease out their separate
contributions to f (ψ). This can easily be done by
utilizing Eq. (4) and taking the
time average of, respectively, the synaptic current Isyn and/or the effective current term gc
VDSout resulting from the electric polarization between the two chambers (see
Appendixes A and B for details on these two terms). In , we have plotted the coupling function for a
neuron with the same parameters as in but with no externally applied electric field. In this case, we have
chosen no applied electric field because we simply want to examine the phase
response of a neuron purely due to the firing of its neighbor within the
resistive array.
are, respectively, the graphs for the coupling function f
(ψ) with and without Isyn. Since the neuron remains embedded within the resistive array, the
firing of the neighboring neuron can still have a modulating effect on its
firing time even when Isyn = 0. However, as one can see from this graph, the
contribution to the coupling function from the ephaptic coupling () is substantially smaller than
the contribution from the synaptic channels (). Nonetheless, since it is the difference
between the coupling functions Δ f
( E, Ψ) that is important in determining the
phase-locked criterion, this small contribution plays a significant role in
determining the synchronization properties of the neurons. Furthermore, it is
important to note that in contrast to the larger but positive-definite
contribution from the excitatory synapes, the ephaptic contribution can be both
positive and negative. Thus, the ephaptic connection has the potential to both
advance and retard the phase of the next spike of the postsynaptic neuron. To illustrate the utility of the reduced phase model in analyzing the
synchronization among our PR neurons, we consider three examples respectively in
the locked in-phase state, the phase drifting state, and the locked anti-phase
state. Our example for the in-phase locked state is from two non-identical
phase-locked neurons. The parameters were E = 600
mV/cm, gc(1) = 2.1
mS/cm2, and gc(2)
= 2.058 mS/cm2. From the direct simulation of this
system, the ensemble has a single locked state with a phase-lag between the two
neurons near Ψ = 5.7 rad (see ). The coupling functions
f1, 2 for both neurons were calculated using
XPPAUT and the results are graphed in . The phase sensitivity curve Δf for
these two coupled neuron is shown in . Note that it crosses the natural frequency mismatch
Δω = 0.5 rad/sec line near its right
positive peak. The stable equilibrium state predicted from this crossing point
roughly agrees with the phase locked value of Ψ = 5.7
rad observed in the full model (). As we have discussed previously, as Δω
increases, the stable equilibrium moves toward the in-phase regime
near Ψ = 0 (or 2π ). Due to
the intrinsic synaptic delays, the locked phase between the two neurons cannot
be exactly at Ψ = 0 (or 2π). This slight
phase lag in the in-phase locked neurons is exactly what we have observed in our
simulation. The next example is for the case when the two neurons do not phase lock
to each other. The parameters were chosen in the asynchronous region with
E = 200 mV/cm, gc(1) = 2.1 mS/cm2, and gc(2) = 1.68 mS/cm2. In this case, the
phases of the two coupled PR neurons drift with respect to each other, as can be
seen in . Again, using XPPAUT,
we have calculated the coupling functions for these two neurons in the reduced
phase model and the phase sensitivity graph Δf is
plotted in . As one can see from
this graph, Δω does not intersect
Δf at all, and the two neurons are not expected to
phase lock. This is what is observed in the actual numerical experiment. The last example is for the anti-phase locked state. As we have seen in
our simplified phase model ()
previously, identical neurons (Δ gc = 0%) with excitatory coupling typically phase
lock in the anti-phase state ( Hansel,
1995). For E = 200 mV/cm and gc(1,2) = 2.1 mS/cm 2,
Δω is zero and Δ f is given
in . As expected, this curve
crosses Δω = 0 near Ψ
= π. However, upon magnification, one can see that
Δ f actually has three zeros in this region with
the left and right zeros being stable and the middle one at ψ
= π being unstable. Therefore, the reduced phase model
predicts that there should be two distinct locking regions near the anti-locked
state. We ran many trials (2000) of our numerical experiment using the full
model of the coupled PR neurons and formed a histogram of the observed locked
relative phases (see ). From
this histogram, we found two main clusters on either sides of Ψ
= π which indicate a multi-stable phase-locked state in
this anti-phase regime. The two peaks of the histogram align closely with the
two predicted locked phases. However, if one examines the histogram closely, one
notices that there is a spread to the two histogram peaks. This indicates
additional structure in the anti-synchronous regime for the two-coupled PR
neurons. We speculate that the ephaptic coupling through the intercellular
medium may play a subtle role in modulating the synchronization between these
two neurons within this multi-stable region. We have shown that excitatory electric fields can have synchronizing and
desynchronizing effects depending upon the heterogeneity of the neurons and the
strength of the applied electric field. For weak excitatory fields, a careful
modulation of the field amplitude might be able to push a network across a
synchronization boundary established by the natural frequency mismatch of the
neurons. Such a prediction is quite testable in laboratory experiments. Inhibitory applied fields were noted to suppress neuronal activity at modest
field strength, but surprisingly to increase activity and to synchronize neurons at
larger field strengths. In recent simulations, we have demonstrated that this effect
is due to dendritic activation, and a reversal of intraneuronal current flow during
initial field application from dendrite-to-soma to soma-to-dendrite (Munyan et al.,
unpublished observations). Again, such a prediction is readily testable in future
experiments. We further employed a phase oscillator formalism to attempt to simplify the
neuronal interactions using phase sensitivity curves for each individual neuron.
Such an analysis is instructive in that it readily permits dissection of the
synaptic and ephaptic coupling effects on phase, and permits predictions regarding
the conditions required for synchronization. Although ephaptic effects are generally
small, there are clear regimes of phase where ephaptic coupling plays an important
role ( Traub, 1985). Since recent experimental
work ( Francis et al., 2003) has demonstrated
the exquisite sensitivity of neurons to weak electric fields, such ephaptic coupling
may have a more important role in neuronal network synchronization than is generally
appreciated. Furthermore, our entire approach, which focuses on polarization
effects, is one that is operative in the weak electric field regime. Stronger
applied fields and their associated currents can induce depolarization block, a
regime where field orientation is no longer relevant ( Bikson et al., 2001). Larger scale simulations will be required in order to begin to replicate
biological effects more accurately. Although we study dimensionally similar
(quasi-1D) topologies in brain slice experiments, a more accurate representation of
the connectivity of excitatory and inhibitory neurons is required before a complete
model of seizure suppression can be constructed. Since the computational complexity
of such networks increases rapidly with size, the prospect of generating qualitative
predictions about synchronization or desynchronization from a reduced phase model
approach is highly attractive. We are grateful to R. Traub for helpful discussions. This work was supported by NIH
grants K02MH01493 (SJS), K25MH01963 (EB), and R01MH50006 (SJS, PS, BJG). Appendix A: The Neuron Model The PR neuron ( Pinsky and Rinzel,
1994) is a lumped two-chamber model with transmembrane potentials for the
somatic and the dendritic chambers governed by the following equations (also see
), The various ionic, synaptic, and leakage currents are defined as: Id and Is are the injected currents into the soma and the dendrite and IDS
in gc( Vdin − Vsin) is the current that flows between the two chambers of the neuron.
Here, gc is the internal conductance between the soma and the dendrite, and
( Vdin − Vsin) is the intracellular voltage difference between the
two chambers. The heterogeneity between the neurons within our network is
adjusted through this internal conductance parameter gc Specifically, gc(1) = 2.1 (mS/cm 2), and  where Δ gc = 0 – 30%. The geometric factors p and (1 −
p) characterize the proportion of area occupied by the soma and
the dendrite, respectively, so that all currents listed above except IDSin, Id, and Is are in units of μA/cm2. In the original PR
neuron, Isyn is also defined as the total current into the
dendrite compartment as are the other injected currents Id and Is. Thus, the synaptic term in the original PR neuron reads Isyn/(1 − p). In our model,
we explicitly assume Isyn to be the synaptic current per unit membrane area of the dendrite so
that the synaptic current term appears as above without the1/(1
− p) factor. In addition to the current balance equations for the transmembrane
potentials, the kinetics of the gating variables for the different ionic
channels is governed by the following equations: where with y = h,
n, s, c, and
q. The specific choices for the different α and β
function listed above are provided in Table
1. The dynamics of the intracellular Ca 2+
concentration is given by and χ(Ca) = min(Ca/250, 1).
Lastly, the weighting functions for the two synaptic conductances (NMDA and
AMPA) are governed by the following differential equations: In these equations, the sum is over all pre-synaptic neurons and
H( x) refers to the Heaviside step
function, i.e., H( x) = 1 if
x ≥ 0 and
H( x) = 0 if x
< 0. All other parameters for the PR neuron used in our model are
summarized in Table 2. Appendix B. Resistive Array To model the electric field interaction among neurons within the
extracellular medium, we embedded two (or more) PR neurons within a resistive
array as shown in . Although we
have only shown a pair of PR neurons in this graph, a natural extension to a
larger number of neurons in a quasi-one-dimensional array can easily be done by
repeating the single neuron unit horizontally along the chain (Munyan et al.,
unpublished data). In the following three subsections, we describe in detail the
modifications needed to implement this electric field interaction for two PR
neurons embedded within the array. B.1. Ephaptic Coupling Through the Resistive Array is a circuit diagram
for the resistive array with two embedded PR neurons. In this figure, the
bulk resistors are labeled by ‘R’ and the equivalent
terminators are labeled as ‘Z’. (Please see the next
subsection for a discussion of how these terminal resistances were chosen to
model an infinite resistive array.) An externally applied electric field is
modeled by the potential difference Vapp imposed between the top and bottom plates of the array. The neural
activities of the two neurons couple into the array as currents passing into
the four connection points indicated in the figure (see the small open
circles in ). Consider the
top-left connection point in ,
where all the currents of Eq.
(A1) (including the capacitive current) pass from the dendrite
chamber into the array. By applying Kirchhoff’s rules at all
junctions and closed loops within the resistive array at a particular
instant of time, one can write down a linear matrix equation relating the
resistor values within the array and the inputs to the array. In particular,
these inputs are Vapp and the currents from the neurons. The solution of this linear
matrix equation gives the instantaneous current/voltage values along all
branches of the array. We are particularly interested in the voltage
difference VDSout = Vd
out − Vsout between the extracellular nodes where the dendrite and soma are
connected to the resistive array. This voltage difference induces a
polarization between the two chambers of the neurons. In particular, note
that the current IDSin which flows through the neuron’s internal conductance
connecting the two chambers is modulated by VDSout. Without the resistive array, we have IDSin gc( Vdin − Vsin) = gc( Vd − Vs ) as in the original Pinsky-Rinzel model. For a neuron embedded
within the resistive array, we have since the extracelluar nodes outside the two chambers are no longer
at the same potential and VDSout is non-zero (Note that, by definition, we have Vd,s = Vd,sin − Vd,sout). Algorithmically, we solve a linear matrix equation at each time
step to calculate VDSout in terms of the instantaneous activity of both
neurons, i.e., VDSout is a linear function of VD and VS from both neurons. Then, by Eq. (A2), VDSout affects the transmembrane potentials VD and VS in the next time step through the Pinsky-Rinzel
equations Eq. (A1). B.2. Equivalent Resistances for an Extended Infinite Array The resistive lattice which models the extracellular medium is set
up effectively as an array of infinite extent by the use of
equivalent terminal resistors. To obtain these
equivalent terminators, one can consider the two circuits given in . As in our previous disscusion,
the bulk resistors are labeled by ‘R’ and the
equivalent terminators on the right side of the array are labeled
‘Z’. (Although our discussion
is for the terminators on the right side the array, the left equivalent
terminators can be obtained through a similar argument.) If the terminal
resistances are chosen correctly, then for given voltage inputs (V1 and V2),
the currents (i1 and i2) flowing through
the top circuit () should
be equal to the currents ( j1 and j2)
flowing through the bottom circuit (). For this particular arrangement of resistors, a simple way to solve
for these equivalent terminators is to assume that the bulk resistors and
the terminal resistors are proportional to each other as shown in with two free parameters
a and b. Then, by imposing the
condition that i1 = j1 and
i2 = j2 as mentioned
above, one can solve for the required proportionality a and
b. In our numerical simulation, a
= 0.0756826 and b = 2.38394 were
used. Other resistance values employed in our resistive network array are
given in Table 3. B.3. Relation between Extracellular Postassium Concentration
([K+]o) and
the Resistive Network In designing this resistive network, the relation between the
extracelluar potassium concentration
[K+]o and the resistivity in the extracellular medium has also been
taken into account. Choices in
[K+]o can be adjusted through the resistances chosen in our resistive
network. Physiologically, increasing
[K+]o increases cell swelling, which in turn decreases the extracellular
volume space as well as the cross-sectional area of the extracellular
medium. Since the cross-sectional area of the extracellular medium and its
resistance are reciprocally related, the extracellular resistance value
RDSout should increase with elevated levels of extracellular
[K+]o. One can model this dependence by the following linear relation, where RDSin is the intracellular resistance and is inversely proportional to
gc. Thus, as [ K+] o is increased from 3.5 (mM) to 8.5 (mM), the ratio
r(= RDSout
/ RDSin) increases from 0.1 to 0.15. With a given fixed value of RDSin, this implies an increase in the extracellular resistance value
RDSout from ~8(MΩ) to ~12(MΩ), an increase of
approximately 33%. This variation in extracellular resistance is
consistent with experimental measurements of extracellular volume in terms
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