A computational study of background

1
A computational study of background-induced flicker enhancement and feedback mechanisms in
vertebrate outer retina: temporal properties
Steven M. Baer1, Shaojie Chang1,4, Sharon M. Crook1,2, Carl L. Gardner1, Jeremiah R. Jones1,5,
Ralph F. Nelson3, Christian Ringhofer1, and Dritan Zela1
1
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287
2
School of Life Sciences, Arizona State University, Tempe, AZ 85287
3
Neural Circuits Unit, Basic Neuroscience Program, NINDS, NIH, Bethesda, MD 20892
4
The High School Affiliated to Beijing Normal University, Beijing, P.R.China 100052
5
Chemical Sciences , CHAP , Lawrence Berkeley National Lab, Berkeley, CA 94720
Running head: Temporal properties of flicker enhancement and feedback mechanisms
Contact information:
Steven M. Baer, School of Mathematical and Statistical Sciences, Arizona State University,
Tempe, Arizona 85287-1804;
Telephone: (480) 965-1057. Facsimile: (480) 965-8119. e-mail: steven.baer@asu.edu
2
ABSTRACT
In vertebrate outer retina, changes in the membrane potential of horizontal cells affect the
calcium influx and glutamate release of cone photoreceptors via a negative feedback mechanism.
This feedback has a number of important physiological consequences. One is called backgroundinduced flicker enhancement in which the onset of dim background enhances the center flicker
response of horizontal cells. The underlying mechanism for the feedback is unclear but
competing hypotheses have been proposed. One is the GABA hypothesis, which states that the
inhibitory neurotransmitter GABA, released from horizontal cells, mediates the feedback by
blocking calcium channels. Another is the ephaptic hypothesis, which contends that calcium
entry is regulated by changes in the electrical potential within the intersynaptic space between
cones and horizontal cells. In this study, a continuum spine model of cone-horizontal cell
synaptic circuitry is formulated. This model, a partial differential equation system, incorporates
both the GABA and ephaptic feedback mechanisms. Simulation results, in comparison with
experiments, indicate that the ephaptic mechanism is necessary in order for the model to capture
the major temporal dynamics of background-induced flicker enhancement. As a corollary we
have discovered that biasing the cone synaptic voltage directly with surround current can trigger
enhancement, provided a horizontal cell feedback mechanism is attached to the cone. Our
results indicate that the GABA mechanism may play a secondary modulation role. Simulations
that included both mechanisms best fit (qualitatively) enhancement data for cat retina, and
provided valuable insights into interpreting the dynamics of calcium activation in feedback
experiments on goldfish retina.
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INTRODUCTION
The retina is part of the central nervous system and an ideal region for studying information
processing in the brain. It is accessible, well documented, and the subject of research spanning
the clinical, experimental, and theoretical sciences.
Visual processing begins in the outer
plexiform layer of the retina, where bipolar, horizontal, and photoreceptor cells interact. In this
first layer of the visual pathway, knowledge of synaptic feedback mechanisms allows for the
formulation of computational models that encapsulate essential phenomenology. From the
biologist's perspective, mathematical modeling allows one to isolate the functional effects of
various circuitry elements, similar to performing experiments with selective pharmacological
agents or genetic modifications, except that not all the agents or modifications one might like
actually exist. From the mathematician's perspective, the availability of electrophysiological,
anatomical, and molecular data provides a rare opportunity for the construction of complex
multiscale models for the system. In this paper we model, in detail, the subcircuits of the outer
plexiform layer, capturing the temporal dynamics on two spatial scales - that of an individual
synapse and of the receptive field (hundreds to thousands of synapses). We apply the model to
compare theory with experiment for background-induced flicker enhancement in cat retinal
horizontal cells.
In the outer plexiform layer of the retina there are four types of neurons that interact: rods,
cones, bipolar, and horizontal cells. The rods and cones serve as receptor cells, providing the
synaptic input to this layer; whereas the bipolar cells are the output neurons that carry visual
information to the inner plexiform layer. The horizontal cells extend their dendritic branches
widely in the outer plexiform layer and are confined to this layer. A prominent feature of the
4
horizontal cells is their electrical (gap) junctions. The gap junctions are so numerous that it is
reasonable to view the collection of horizontal cells as a syncytium or a conductive sheet. The
horizontal cell syncytium increases the size of the receptive field and mediates interactions
between rods and cones (Frumkes and Eysteinsson, 1988). The horizontal cells in this study are
of the A-type, which is axonless and has a dendritic tree with its terminals ending in cone
pedicles (Kolb 1974; Perlman et al. 2012). Although an A-type horizontal cell does not have
direct synaptic contact with rods, it receives rod inputs indirectly from cones since rod signals
can enter cones via rod-cone gap junctions (Raviola and Gilula 1973; Kolb 1977; Nelson 1977;
Smith et al. 1986; Wu 1994; Schneeweis and Schnapf 1995).
To understand the neural subcircuits within the first layer of the visual pathway, it is important
to draw upon knowledge from both psychophysics and electrophysiology. In human
psychophysics, it is well known that after a brilliant desensitizing flash, cone flicker sensitivity
first increases but then, paradoxically, decreases with a time course paralleling rod dark
adaptations (Goldberg et al. 1983). This interaction between rods and cones is called suppressive
rod-cone interaction (SRCI). Analogous physiological effects involving rod and cone signals
occur in horizontal and bipolar cells (Eysteinsson and Frumkes 1989; Frumkes and Eysteinsson
1988; Pflug et al. 1990). For example, in cat, the onset of dim backgrounds can enhance smallspot flicker responses of retinal horizontal cells (Pflug et al. 1990). This is called backgroundinduced flicker enhancement.
It is generally accepted that background-induced flicker
enhancement is due to horizontal-cone feedback: hyperpolarized horizontal cell dendritic
terminals effectively increase the entry of calcium into the cone terminal, stimulating the release
of transmitter glutamate by the cone presynaptic apparatus. The result is a depolarization of the
horizontal cell; that is, an increase in synaptic gain.
This explanation accounts for how
5
peripheral rod-induced, horizontal cell hyperpolarizations, conducted centripetally to the
horizontal cell dendritic terminals via gap junctions, can enhance postsynaptic cone responses
(Nelson et al. 1990).
Currently there are three hypotheses to explain an increase in calcium entry in response to
horizontal cell hyperpolarization: the GABA, ephaptic, and pH hypotheses. Gammaaminobutyric acid (GABA) is an inhibitory neurotransmitter that occurs in many different
varieties of amacrine cells and in one or more classes of horizontal cell in most vertebrate retinas
(Marc et al. 1995). Under the GABA hypothesis hyperpolarization of the horizontal cell reduces
the release of blocking agent GABA from the horizontal cell, allowing calcium to enter the cone
terminal, stimulating glutamate release by the cone presynaptic apparatus (Nelson et al. 1990;
Corey et al. 1984; Belgum and Copenhagen 1988). A schematic diagram of this hypothesis is
shown in Fig. 9 of Nelson et al. (1990). The other two hypotheses, ephaptic and pH, both
modulate calcium channels on the cone terminal for calcium entry but their mechanisms differ.
The ephaptic hypothesis (Byzov and Shurabura 1986; Kamermans et al. 2001) involves
horizontal cell hemichannels and cone voltage-gated calcium channels. A hemichannel (also
called connexons) is half of a gap junction; that is, it is a gap junction on one cell (here, a
horizontal cell) but not on the adjacent cell, leaving it open into the extracellular space (Spray
2006). According to the ephaptic hypothesis, as depicted in Fig. 1A of Fahrenhort et al. (2009),
hyperpolarization of the horizontal cell increases hemichannel-mediated current flow into the
horizontal cell. Since this current must pass through an intersynaptic resistance, the potential
drop over this resistance will increase, making the potential in the synapse even more negative.
The voltage sensitive calcium channels on the cone terminal will sense a larger depolarization,
opening more calcium channels and therefore more calcium influx into the cone which
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stimulates glutamate release. In other words, the current flow into the hemichannel coupled with
the high resistance path of the synaptic cleft shifts the calcium current activation in the cones to
more negative potentials (Kamermans and Fahrenfort 2004; Fahrenfort et al. 2009).
A
compartment model developed by Fahrenfort et al (2009) supports the ephaptic hypothesis. This
model was developed as a rebuttal to the model by Dmitriev and Mangel (2006) that claimed it is
unlikely that an electrical mechanism plays a significant functional role in the feedback response.
A recent computational study at the single synapse level, using a drift-diffusion (Poisson-NernstPlank) model, reproduced experimental current-voltage curves for the goldfish retina in response
to a bright spot with and without an illuminated background, adding support to the ephaptic
hypothesis (Gardner et al. 2012).
The pH hypothesis provides an alternative to the ephaptic mechanism. Under this hypothesis
extracellular pH changes in the intersynaptic cleft mediate the feedback. Specifically, horizontal
cell hyperpolarization alkalinizes the cleft, and the decrease in extracellular proton concentration
then alters the gating of pH-sensitive calcium channels in the presynaptic cone pedicle and
results in an increase of cone calcium current in the form of a negative shift of the current’s
activation potential (Barnes et al. 1993, Hirasawa and Kaneko 2003). The mechanism by which
horizontal cell polarization modulates the extracellular pH is still unknown, although
hemichannel-mediated proton transport (Kamermans and Fahrenfort 2004, Fahrenfort et al.
2009), amiloride-sensitive proton channels (Vessey et al. 2005), or some other proton transport
system (Bouvier et al. 1992, Hirasawa and Kaneko 2003) on the horizontal cell spines serve as
possible candidates.
In 1990, Nelson and Pflug (Pflug et al. 1990; Nelson et al. 1990) recorded from horizontal
cells of cat retina and demonstrated that photopic flicker components of horizontal cell responses
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were enhanced during hyperpolarization of the horizontal cell by background light (see Fig. 1 in
Pflug et al. 1990). Figure 1A shows the enhancement effect from simulations. To quantify the
enhancement
effect,
a
variable
E
or
“percent
enhancement”
was
defined
as
E = 100 [(Fbkgd / Fdark) – 1], where Fbkgd and Fdark are mean flicker response amplitudes for
background illuminated and dark, respectively (Frumkes and Eysteinsson 1988). Nelson and
Pflug's experiments revealed some important dynamical properties associated with the feedback
mechanism. One was a slow repolarization of background-induced hyperpolarization, called
‘sag’ or ‘rollback,’ followed by an overshoot or post-inhibitory rebound when the background
light was turned off (Fig. 1A and Fig. 1 in Plug et al. 1990). Another was the two-limbed nature
of the E vs. flicker frequency curve: E gradually increased below 20 Hz and then rapidly
increased above this frequency (Fig. 1B and Fig. 4 in Pflug et al. 1990). In addition, Nelson and
Pflug observed background-induced changes in flicker waveform in the form of a phase shift.
They found a phase shift in flicker waveform between a flicker cycle in the dark and a flicker
cycle with background illuminated (Fig. 1C and Fig. 5 in Pflug et al. 1990).
The dynamical properties associated with background-induced feedback experiments provide
important clues for determining if one or more of the competing hypotheses for calcium entry are
appropriate. In this paper we focus on GABA and ephaptic mechanisms. We formulate and
analyze a partial differential equation system that models the horizontal cell syncytium and its
numerous spiny processes forming synapses within invaginated synaptic clefts of the cone
pedicles, and compare simulation results with enhancement data for slit and square test regions.
In addition, we consider the case of a constant (non-flickering) circular or disk test region to
study steady state calcium current-voltage relationships in the presence and absence of a full
field background illumination.
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This study demonstrates that an ephaptic mechanism produces reasonable qualitative
agreement with temporal data from background-induced flicker enhancement experiments. In
addition we find that although GABA produces enhancement, this mechanism alone is
insufficient to reproduce the full repertoire of temporal dynamics exhibited by the flicker
enhancement experiments. We find that the best fit to the enhancement data is obtained when
both the GABA and the ephaptic mechanisms are included in the model. Our simulations show
that for this hybrid model GABA effectively fine tunes the more dominant ephaptic mechanism.
It is important to state that this study does not rule out the pH hypothesis as a possible
mechanism. It is indeed possible that a pH mechanism could produce similar computational
results as our hybrid model. Finally, we reserve for future study the spatial properties of the
enhancement effect; for example, comparing the dependence of percent flicker enhancement on
test region size for different geometries. This will require a modification of the model to account
for the desensitization of rods when exposed to bright flicker (see Discussion).
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METHODS
In this section we formulate a mathematical model of the horizontal cell syncytium and its
numerous processes that form synapses within cone pedicles. This multiscale model incorporates
both GABA and ephaptic synaptic mechanisms and captures dynamics on two spatial scales: the
scale of the individual synapse and the scale of the receptive field.
The horizontal cell network and spine processes
We model the horizontal cell network or syncytium as a thin conducting sheet in the xy-plane
(Lamb, 1976; Nelson 1977; Nelson et al. 1990). We assume that the sheet or slab of horizontal
cells constitutes a planar region that includes the origin of the xy-plane (0,0). In addition,
emanating from the sheet are numerous horizontal-cell dendritic processes (spines) that form
synapses within invaginations of cone pedicles (Winslow et al. 1989). The spine density N is
defined as the number of spines per unit physical area. Over an area element ∆A, the spines
deliver current ∆A N ISS to the syncytium, where ISS represents the current flowing through an
individual spine stem. This stem current is expressed as an I∙R voltage drop across the spine
stem resistance RSS (MΩ), given by
I SS 
U H  VH
RSS
,
(1)
where UH and VH (mV) denote the membrane potentials in the horizontal cell spine head and
network, respectively. The spine stem is modeled, as in previous studies, as a lumped Ohmic
resistor, neglecting the stem's membrane and cable properties (Miller et al. 1985; Rall and Segev
1987; Segev and Rall 1988; Baer and Rinzel 1991).
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The equation for the membrane potential in the sheet VH (x,y,t) is a current balance relation
for the capacitive, gap junction, stem, and ionic currents given by
Cm
VH
1 2

 VH  N I SS  I ion ,
t
RS
(2)
where Cm (F/cm2) is the specific membrane capacitance, RS (M) is the horizontal cell gap
junction sheet resistance, and Iion (μA/cm2) is an ionic current density passing through the
horizontal cell membrane.
Multiplying through by Rm (∙cm2), the passive membrane
resistance for horizontal cells, we substitute the membrane time constant m = Rm Cm, the length
constant   Rm / RS (Naka and Rushton 1967), and the dimensionless spine density n   2 N
into Eq. (2) to arrive at the equation for the horizontal cell syncytium
m
VH
  22VH  RS n I SS  I ion Rm .
t
(3)
In this study we assume that the spine density is constant. To justify the use of the above
continuum spine equation for horizontal cells, the spine density must be sufficiently large (Baer
and Rinzel 1991). A typical mammalian cone pedicle has approximately 30 triads of processes
(Ahnelt et al. 1990). Each triad typically contains one bipolar and two horizontal cell dendritic
terminals (Kolb et al. 2012). Therefore, a cone pedicle makes synaptic contacts with about 60
horizontal cell spines. For cat retina, a modest estimate of cone density is approximately 6.4 ×
105
cones/cm2 (Steinberg et al. 1973; Williams et al. 1993), and estimating an average of 60
spines/cone yields a spine density of
N ≈ 384 × 105 spines/cm2. For cat, reasonable estimates
are Rm = 104 ∙cm2 and RS = 12 M. With these values,  ≈ 289 μm and the number of spines
11
per electrotonic area is therefore n ≈ 3.2 × 104, which is large enough to justify a continuum
formulation.
The spine stem resistance values are also quite large. Modeling the spine stems as circular
cylinders of diameter dss and length Lss, and with constant intracellular resistivity Ri, the spine
stem resistance is defined as RSS = (4Lss Ri ) / ( dss2 ) . In cat, Ri is estimated to be about 200 Ω∙cm
(Smith 1995; Spruston and Johnston 1992), Lss ≈ 5 μm and dss ≈ 0.1 μm. For these estimates,
RSS
≈
1300 M, which is quite high considering we are not accounting for possible
constrictions in the stem that would raise the stem resistance even higher. It is important to note
that there are no excitable channels in this system capable of generating action potentials.
Consequently the spine head potentials closely approximate the slab potential (see Discussion).
The above estimates for spine density and stem resistance are summarized in Fig.2.
We model the horizontal cell spine head as an isopotential compartment with surface area
Ash (μm2) and specific membrane capacitance Cm, thus individual spines have a capacitance of
Csh = Ash Cm (μF) . The equation for the membrane potential UH (x,y,t) in a single horizontal cell
spine head is obtained from a current balance relation for the capacitive, ionic, and spine stem
currents given by
Csh
U H
  Ash I ion  I syn  I ss .
t
(4)
The synaptic current Isyn, which is activated by transmitter glutamate [GL] (M) released from
the cone, is defined by
I syn  g syn U H ,
(5)
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where gsyn = ksyn [GL] (nS) is the synaptic conductance. Here the synaptic reversal potential is
zero, so that the glutamate current is inward when the cell is hyperpolarized. The glutamate
concentration in the synaptic cleft is governed by
 GL
[GL]
 kCa I Ca  [GL] .
t
(6)
The time rate of change of glutamate is balanced by the release of glutamate triggered by cone
calcium current (ICa < 0) and the uptake of glutamate by transporters.
In the present model we assume that ‘sag’ or ‘rollback’ of the horizontal cell membrane
potential, followed by post-inhibitory rebound, is due to ionic currents. In fish and mammalian
retinal horizontal cells, T-type or T-type-like calcium currents were reported (Löhrke and
Hofmann 1994; Pfeiffer-Linn and Lasater 1993; Schubert et al. 2006; Sullivan and Lasater
1992), together with N-type, L-type and other types of calcium currents, sodium current, and
various types of potassium currents (Löhrke and Hofmann 1994; Ueda et al. 1992). Therefore,
we model Iion in Eqs. (3) and (4) as the sum of a nonlinear inward ‘sag’ current and a linear
leakage current. Here our modeling follows the form of Golomb et al. (1994). Specifically,
Iion (V , h)  (V  Esag ) gsag h  (V  ELH ) g LH ,
(7)
where h, an inactivating gating variable, is defined as
h
h
 h (V )  h .
t
(8)
Here,
h (V )  (1  e(V h )/ h )1
,
(9)
where τh, θh, and σh are constants. In the above equations V and h are replaced by VH and hV
in Eq. (3) and by UH and hU in Eq. (4) .
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The cone
The horizontal cell spine head forms a synapse with the cone pedicle. We assume that within
an invagination of the cone, the adjacent cone membrane area is of the same magnitude as the
surface area of the horizontal cell spine head. The part of the cone forming the invagination is
treated as an isopotential compartment, and across its membrane the capacitive current is
balanced by a leakage current, a “dark” current, and two input currents. The cone equation is
Cm
VC
  g LC (VC  ELC )  I dark  I flick  I bkgd ,
t
(10)
where the cone membrane potential is denoted by VC(x,y,t). The first term on the right of Eq.(10)
is a leakage current density with reversal potential ELC (mV) and conductance gLC (nS/cm2). We
note that calcium entry into cones activates chloride currents (Lansansky 1984). The activation
of chloride currents occur on a relatively long time scale, between 200 to 2000 ms. Rather than
adding another linear term for the chloride current in Eq.(10), we lump the chloride current into
this leak term. The second term Idark (μA/cm2) is the density of a constant “dark” current
representing a steady inward cation current mainly contributed by sodium ions that depolarize
the membrane potential in darkness (Hagins et al. 1970; Bear et al. 2007; Yau 1994). In this
paper there are two applied current sources Iflick and Ibkgd (μA/cm2). The current density Iflick
(μA/cm2) represents current transduced from a flicker stimulus, applied over a region S centered
at the origin in the xy-plane. The current density Ibkgd represents current transduced from a
constant background stimulus. In order to compare computational results to experimental data,
we modeled both slit and square test stimuli. In addition, we modeled disk (circular) test stimuli
and found that the results were nearly indistinguishable from square test stimuli of similar size,
but the disk model had the added advantage of being computationally more efficient.
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It is important to point out that A-type horizontal cells, which are connected only to cones,
respond with a mixture of rod and cone signals. This mixing is thought to arise at the
photoreceptor level where rods and cones are interconnected by gap junctions. In addition,
feedback to horizontal cells from the interplexiform cells has not been observed in cat retina
(Pflug et al. 1990; Kolb and West 1977). Therefore the current Ibkgd, representing the diffuse
background stimulus, is modeled as entering the cone through rod-cone gap junctions, whereas
Iflick is modeled as resulting from the cone transduction process.
Rods within the test region are light adapted by the flicker stimulation and therefore the rod
signal within the test region is significantly reduced. In addition, and independently, rods
hyperpolarized by the background may act globally to shorten the horizontal-cell length constant
by uncoupling gap junctions (Nelson et al. 1990). A more general model that includes an
additional rod-equation and a state-dependent gap junction resistance RS would be required to
adequately model the above two effects (see Discussion). However, our focus in this paper is on
temporal properties, subject to a few changes in test region size and geometry. Therefore, to
compensate for the above effects we reduce the background current within the stimulus region to
 Ibkgd , where 0 ≤  ≤ 1. A smaller value of γ implies a larger desensitization of rods within the
region. In our simulations we will change the value of , which we will call the desensitization
factor, only when the flicker stimulus size, geometry, or irradiance is changed.
Finally, we assume that the influx of calcium current into the cone has a negligible effect on
the cone membrane potential, therefore Eq.(10) has no term for ICa feedback. The effect of ICa
on cone membrane potential is assumed to be small since the calcium conductance is small
relative to the total membrane conductance of the cone (Kraiaij et al. 2000; Verweij et al. 1996),
and under physiological conditions, the depolarizing effect of ICa on cone membrane potential is
15
counteracted by the hyperpolarizing effect of calcium-activated chloride currents (Kamermans
and Spekreijse, 1999; Kraaij et al. 2000; Verweij et al. 1996). Under these assumptions the cone
potential VC , computed explicitly from Eq.(10), drives (but is not driven by) the rest of the
system.
Figure 3 illustrates the applied currents Iflick and Ibkgd in Eq.(10), defined for the slit test-region
S shown in Fig.3A. In general, S is a simply-connected region in the plane, centered at the
origin. In this study we look at the flicker response of the horizontal cell membrane potential at
the center of square, slit, and disk test-regions. Figure 3B shows the “square” wave flicker
stimulus Iflick as a function of time. However, as the insets illustrate, the onset and offset of each
wave cycle is defined to be smooth and sigmoidal. Likewise, the spatial transition for both Iflick
and Ibkgd across the boundary of S is sigmoidal, as shown in the Fig.3C insets for the minimum
values of Iflick.
Finally, we reduce the background current inside the stimulus region
by
multiplying Ibkgd inside S by the fraction γ as shown in Fig.3D. Note that the offset speed of Ibkgd
is set slower than its onset speed to account for rod after-effect, namely, the slow return of the
rod-induced hyperpolarization towards its baseline after the background is turned off (Steinberg
1969a, 1969b; Pflug et al. 1990). The functions used to generate Iflick and Ibkgd for slit, square,
and disk test-regions are explicitly defined in the APPENDIX.
The horizontal cell-to-cone feedback synapse
We model the cone calcium current ICa (pA) as
 Ca
I Ca
(VC  U H  ECa ) gCa

 I Ca .
t
(1  e (VC U H  A)/ B ) (1  kOCa [G])
(11)
16
Here,
ICa
is
inward
ICa  0
if
and
outward
if
ICa  0 .
The
factor
(VC  U H  ECa ) gCa (1  e(VC U H  A) B ) is a linear current multiplied by a sigmoidal activation
curve (Barnes et al. 1993). In addition, the dependence on UH, scaled by , phenomenologically
models the ephaptic mechanism (see Discussion) and is motivated by the experimental results on
goldfish retina (Verweij et al. 1996).
The expression 1/(1+kOCa[G]) is derived from a Hill
function on the binding kinetics of GABA to open calcium channels, and represents the fraction
of open calcium channels as a function of extracellular GABA concentration [G] with
kOCa (1/M) being the binding constant. Thus, the product of the maximum conductance gca and
the fraction 1/(1+kOCa[G]) reflects the GABA-mediated modulation of cone calcium
conductance. The last term represents calcium removal from the cone by extrusion to
extracellular space or binding to intracellular stores. A novel feature of our model is that Eq.(11)
allows for the option of a hybrid mechanism combining both GABA and ephaptic mechanisms.
We explore the possibility that both mechanisms jointly contribute to the feedback effect.
We model the dynamics of [G] as
G
[G]
 kG (U H  EG ) ,
t
(12)
where
EG 
RT  ln([G ] [Gi ]) 


F 
ni

 log ([G ] [Gi ]) 
RT

ln(10)  10

F
ni


is the Nernst potential for GABA.
In Eq.(12),
G
(13)
and kG are the time constant and scaling
factor for the GABA kinetics. Based on the fact that the release of GABA is voltage-dependent
17
(Schwartz 1982, Schwartz 1987), GABA is treated as an ni-charged particle whose balance
between its extracellular concentration [G] and intracellular concentration [Gi] is controlled by
the difference between UH, the membrane potential of the horizontal-cell spine head, and EG, the
Nernst-type ‘reversal potential’ of GABA. Equations (12) and (13) are for reverse transport of
GABA (i.e., non-vesicular GABA release) since they model both the GABA release from and
reuptake into horizontal cells as membrane transport processes (Schwartz 2002). There is also
some evidence for vesicular release of neurotransmitters from horizontal cells (Hirano et al.
2007). The GABA release mechanism modeled here is not vesicular. In Eq.(13), R is the gas
constant (8.314 J mol-1 K-1), T is the temperature (in K), and F is the Faraday constant (96485 C
mol-1). In our simulations we use the same temperature as in the enhancement experiments,
namely 37oC, corresponding to (RT/F) ln(10) = 61.5 mV.
Summary of the equations for a general two-dimensional test region
Substituting Eq. (1) into (3), Eqs. (1) and (5) into (4) (with gsyn = ksyn [GL]), Eq. (13) into
(12), and including the equations associated with Iion for the slab and spine heads (Eqs. (7)-(9)),
the continuum model for a two-dimensional horizontal cell sheet with n spines per electrotonic
area is
m
VH
U  VH
  22VH  RS n H
 I ion (VH , hV ) Rm
t
RSS
(14)
Csh
U H
U -V
 ksyn [GL] U H  H H  Iion (U H , hU ) Ash
t
RSS
(15)
 GL
[GL]
 kCa I Ca  [GL]
t
(16)
18
I Ca
(VC  U H  ECa ) gCa

 I Ca
t
(1  e (VC U H  A)/ B ) (1  kOCa [G])
 Ca
G
h
(17)

[G]
RT  ln([G] [Gi ])  
 kG  U H 

 
t
F 
ni


hV
h
 h (VH )  hV ,  h U  h (U H )  hU ,
t
t
(18)
(19, 20)
where VC is the solution to the equation
Cm
VC
  g LC (VC  ELC )  I dark  I flick  I bkgd .
t
(21)
In the above equations, the state variables depend on x, y, and t. Functions Iion and h∞ are
defined by Eqs. (7) and (9) respectively. Parameter values for Eqs. (14)-(21) are listed in Table 1.
In the absence of Iflick and Ibkgd the steady-state values of the state variables are chosen as the
initial conditions and are listed in Table 2. Reference parameters for Iflick and Ibkgd, are listed in
Table 3. The overall dynamics and feedback mechanisms for this model are summarized in
Fig.4.
In general, the equations for all geometries are solved by the numerical method of lines.
Specifically, the partial differential equation system (14)-(21) is defined over a finite domain and
then discretized in space using centered differencing for the spatial derivatives. This generates a
system of ordinary differential equations (ODEs) for each grid point, which is then solved using
MATLAB’s built-in stiff ODE solver ode23tb (Mathworks 2012), an L-stable implicit RungeKutta method implementing the TR-BDF2 algorithm (Bank et al. 1985). The method is second
order accurate in both space and time (LeVeque 2007). The computation time is significantly
reduced by first taking advantage of symmetries inherent in slit, square, and disk geometries.
19
These symmetry reductions, the boundary conditions for the finite domain problem, and the
specific functions used to generate Iflick and Ibkgd are presented in the APPENDIX.
20
RESULTS
Our objective is to determine which hypothesis, GABA or ephaptic, best fits the temporal
enhancement data. To address this we have formulated a partial differential equation system
(14)-(21) that models the electro-chemical dynamics of the horizontal cell syncytium and its
numerous processes that form synapses within cone pedicles. We now compare simulation
results with enhancement data for square and slit test-stimulus regions. In addition, we consider
the case of a constant (non-flickering) circular or disk shaped test-region to compare feedbackinduced shifts in calcium current for the competing hypotheses.
Background-induced flicker enhancement
Figure 5 simulates the experiment (Pflug et al. 1990; Fig. 1) where a low amplitude, 16 Hz,
250 m flickering red square test-stimulus is followed by a broad-field, blue diffuse background
stimulus of duration approximately one second. In both the experiment and simulations the time
course of the horizontal cell membrane potential VH is recorded at the center of the test region.
In the experiment, the application of the background stimulus induces a hyperpolarizaton of
about 5 mV. In addition there is an enhancement effect in which the amplitude of the oscillatory
response nearly doubles. Recall that the enhancement effect is quantified by the percent change
in the mean flicker response amplitude, given by
E  100
( Fbkgd  Fdark )
Fdark
,
(22)
where Fbkgd and Fdark are the mean flicker response amplitudes for when the background is
illuminated and dark respectively. In Eq. (22), a flicker response amplitude that doubles when
the background is on (Fbkgd = 2 Fdark ) has a percent enhancement of E = 100. If E > 100 then
21
Fbkgd is more than twice the value of Fdark , E = 0 implies no change in mean flicker response
amplitude, and E < 0 corresponds to a decrease in Fbkgd when compared to Fdark . Experiments
have shown that the percent enhancement depends on the flicker frequency, the geometry and
size of the stimulus region over which it is applied, as well as the magnitude of the diffuse
background stimulus.
Figure 5A, which is a re-scaling of the plot shown in Fig. 1A, is qualitatively similar to the
experimental result in Fig.1 of Pflug et al. (1990) where the flicker response amplitude nearly
doubles when the background is on. Here we simulate a hybrid case where both the ephaptic and
GABA mechanisms are included. To accomplish this we set kG = 1 in Eq. (18) for the GABA
concentration and  = 0.88 in Eq. (17) for the calcium current. In addition to the enhancement
effect the simulation captures the ‘sag’ and post-inhibitory rebound dynamics seen in
experiments. However, in our model the enhancement effect is independent of the sag and postinhibitory dynamics. That is, when Iion and its auxiliary equations are removed from the
equation set, enhancement not only persists but retains its magnitude (see Discussion).
The beauty of doing simulations is that we can selectively block the ephaptic mechanism, the
GABA mechanism, or both. The GABA mechanism is blocked by setting kG = 0 in Eq. (18),
thereby forcing the GABA concentration to remain constant. This is the ephaptic-only case
shown in Fig. 5B. Here the dynamics are similar to Fig.5A but the percent enhancement is
reduced from 99 to 87. To block the ephaptic mechanism, GABA is reinstated by re-setting
kG = 1, but now the ephaptic mechanism is turned off by setting  = 0 in Eq. (17). In the
ephaptic feedback mechanism, hyperpolarization of the horizontal cell causes calcium channels
to open. Setting  = 0 in Eq. (17) removes the dependence of the calcium current on the
horizontal cell potential UH. The only way for calcium channels to open is by changes in the
22
cone potential VC.
The result for the GABA-only case, shown in Fig.5C, is a large
hyperpolarization accompanied by a sharp reduction in percent enhancement to 22. In Fig. 5D,
both mechanisms, GABA and ephaptic, are selectively blocked (kG = 0, The dynamics are
similar to Fig. 5C, but the percent enhancement becomes negative (E = -0.75) which means that
the flicker response amplitude decreases when the background is illuminated.
Figure 6 simulates a background-induced flicker enhancement experiment for a 16Hz, 250m
slit test-stimulus. Although we do not have experimental results showing the time course of the
flicker dynamics, the experimental value for the percent enhancement for this frequency is
approximately 40 (Pflug et al. 1990; Fig.4). In our simulations, the percent enhancement for
the hybrid mechanism is 37 and for the ephaptic-only case 32. The GABA-only mechanism is
also close to the experimental value with an enhancement of 35. When both mechanisms are
blocked, enhancement persists but falls to 23. This significant but reduced enhancement for the
control case is primarily due to the low γ value as compared to the square stimuli. This result
demonstrates that there are potentially three enhancement mechanisms in the model: ephaptic,
GABA, and Ibkgd biasing (see Discussion). Thus Ibkgd , depending on magnitude, may either
enhance the flicker response, or diminish it.
Frequency and phase properties of background-induced flicker enhancement
Pflug et al. (1990) have shown in experiments that flicker response amplitudes decline with
increasing frequency, but background-induced flicker enhancement increases. In addition, they
reported that although two different cells under different stimulus conditions (e.g.; square versus
slit stimuli) can produce quite different frequency dependencies, the curves reveal a common
feature. The enhancement versus frequency curves have a “two-limbed” feature, characterized
23
by a gradual increase in percent enhancement below 20 Hz and a relatively rapid increase above
this frequency.
Figures 7 and 8 compare simulations with experiment for square and slit test-regions. The
solid curves are the percent enhancement plotted against flicker frequency for each of the
different mechanisms. The dotted curves consist of points taken from the cubic regression of the
experimental data in Fig.4 of Pflug et al. (1990). In panels A and B of both figures the hybrid
and the ephaptic-only cases qualitatively fit the experimental results for both squares and slits,
capturing the “two-limbed” feature. In contrast, panels C and D clearly show that the GABAonly and the control case do not qualitatively fit the data. Here, for both square and slit testregions, the percent enhancement decreases with increasing frequency, and can even become
negative for large frequencies.
Although in experiments enhancement increases with frequency, these same experiments
indicate that flicker response amplitudes decrease. This decrease in amplitude, before and during
background onset, is shown in Fig. 3 of Pflug et al. (1990) for a square test-region. Our
simulations confirm this result for hybrid and ephaptic-only cases with both square and slit testregions. Figure 9 shows the declining flicker response amplitude with increasing frequency for
the hybrid case subject to a square test-stimulus. The time scale in Figure 9 is stretched to
highlight the similarities between the model simulations and Fig. 3 in the experiment. Note that
although the flicker response amplitudes decline with frequency, background-induced flicker
enhancement increases from 60 at 10 Hz to greater than 100 at 20 Hz. As the dark flicker
response amplitudes become negligibly small the enhancement rapidly increases to nearly 300.
However, for these large enhancement values, cone flicker sensitivity most likely decreases due
to the sharp reduction in flicker response amplitude.
24
Another useful way to test the validity of the model and the competing hypotheses is to look
at background-induced phase shifts in flicker enhancement experiments. In Fig. 5 of Pflug et al.
(1990) the single-cycle waveforms of flicker response with and without background illumination
were compared for a square test stimulus. They scaled the dark flicker response amplitude to the
amplitude of the background-on flicker cycle and then superimposed the plots. For higher
frequencies ( > 11 Hz) the repolarizing phase of the flicker cycle with background-on was shifted
approximately 5 ms in advance of the dark flicker cycle. The hyperpolarizing phase was less
affected, with the background-on flicker cycle shifted only slightly in advance of the dark flicker
cycle.
Figures 10 and 11 show the background-induced phase shift in the flicker response VH for the
model with square and slit-shaped test regions. In Fig. 10 both the hybrid and the ephaptic-only
cases produce background-induced phase advances during the repolarizing phase of the
waveform.
However, this phase advance continues into the hyperpolarizing phase with a
magnitude significantly larger than seen in the experiment. The GABA-only case and the
control case produce background-induced phase delays which are not consistent with
experimental observation. Figure 11 shows that the results for a slit-shaped test region are
qualitatively similar to the square test region, but the shifts are significantly smaller in
magnitude. Unfortunately, experimental phase shift data is unavailable for the slit stimulus
geometry.
It is important to note that the specific forms of the phase shifts seen in Figs.10 and 11 are
shaped by the entire system of equations. However, there are two equations in our system that
when isolated, can generate a phase shift response. The first is Eq. (17) for ICa . When UH is
decoupled from Eq. (17), by setting α = 0, and [G] is held constant by setting kG = 0 in
25
Eq. (18), we have the control case where neither feedback mechanism is present, and the cone
potential VC is the only time varying forcing function in Eq. (17). At background-on VC is
prescribed to hyperpolarize without a change in amplitude and without a phase shift. Yet Fig.12
(top left) clearly demonstrates that ICa exhibits a background-induced phase shift. In Eq.(17) the
linear current multiplied by the sigmoidal activation function is, mathematically, the source of
the background-induced phase shift for this case.
Equation (15) is the other equation in the system that can independently generate a phase shift
response; specifically the synaptic term involving glutamate. To demonstrate this we have
decoupled the stem current from Eq. (15) and prescribed [GL] as a sinusoidal forcing function of
the form [GL] = D + A sin ωt , choosing D and A to roughly approximate the [GL] response for
the hybrid case: (ω = 2π/57.5; D = 15; A = 4 for background-off and A = 7 for backgroundon). We found that by simply increasing the amplitude of [GL], corresponding to backgroundon, was sufficient to produce a robust phase shift response in UH. Furthermore, we have found
that when the ionic current was replaced by a linear leakage current, in Eq.(15), the backgroundinduced phase shift persisted. It is important to re-emphasize that although both Eq. (15) and Eq.
(17) have the capacity to generate a phase shift response independently, it is the coupling of the
entire system of equations that ultimately shapes the background-induced phase shifts observed
in Figs. 10 and 11.
Glutamate reuptake can influence background-induced phase shifts.
In Fig.12 the
background-induced phase shifts of ICa , [GL] , and UH are compared for a ten-fold increase in
the glutamate reuptake rate. In the absence of the ephaptic and GABA kinetics (Fig.12A),
increasing glutamate reuptake by a factor of ten has no effect on the phase shift dynamics of
inward calcium current ICa (left panels). The reason is that in Eq. (17) UH is decoupled (α = 0)
26
and [G] is constant, and since [GL] influences ICa through UH and [G] in Eqs. (15) and (18)
respectively, increasing the reuptake rate has no influence on Eq. (17) for this case. Figure 12A
(middle) shows that increasing the reuptake rate of [GL] results in an approximate ten-fold
decrease in [GL] accompanied by a small change in phase shift dynamics (Fig.12A, middle
panels). In contrast Fig.12B for the ephaptic only case exhibits a more pronounced change in the
phase shift dynamics for all three variables due to the re-coupling of UH in Eq. (17). Here ICa
exhibits a change in phase shift dynamics (Fig.12B, left panels) that influences the dynamics of
[GL] in Eq.(16). This contributes to a significant change in the phase shift dynamics of UH
(Fig.12B, right panels).
Background-induced shift in calcium current and center-surround antagonism
Here, we consider the case of a constant (non-flickering) circular or disk shaped test-region in
the presence and absence of a full-field background illumination to compare feedback-induced
shifts in calcium activation for the competing hypotheses. In addition we show that simulations
with the full model, Eqs. (14)-(21), are qualitatively consistent with the results of feedback
experiments on goldfish retina, and that ICa and [GL] exhibit center-surround antagonism.
Recall that the formulation of Eq. (17) for calcium activation was partially phenomenological.
That is, the term UH (UH < 0) was simply inserted into Eq. (17) in order to translate the
steady-state ICa vs. VC
hyperpolarized.
curve to the left when the horizontal cell membrane potential is
This effectively shifts the calcium current to more negative values, or
equivalently, increases the inward calcium current into the cone. This formulation was motivated
by the ephaptic hypothesis proposed by Kamermans and Fahrenfort (2004). Their conceptual
model is based on the experimental results for goldfish data shown in Fig. 11 of Verweij et al.
27
(1996).
In contrast, the modeling of the GABA mechanism, also in Eq. (17), is less
phenomenological since it is based on the Hill equation, used here to characterize the
biochemical release of GABA by the horizontal cells.
The purpose of this section is to show
that our phenomenological approach to model the ephaptic mechanism captures the qualitative
features of the goldfish experiment and adds insight into how a hybrid mechanism may play a
role in the feedback response.
For both the GABA and ephaptic hypotheses, the background-induced hyperpolarization of
horizontal cells shifts the voltage-dependence of the calcium current to more negative values. In
the goldfish experiment, steady-state ICa vs. VC curves are plotted for a constant disk-shaped
bright spot in the presence and absence of a full field background illumination (See Fig.11a in
Verweij et al. 1996). After the introduction of the background illumination, inward cone calcium
current increases.
This results in a slight downward shift to the left of the
I-V curve,
accompanied by a very small upward shift in the right-most branch of the curve.
In Fig. 13B, our simulation of the ephaptic-only case is consistent with a shift to the left.
Note that this shift is approximately a pure translation of the curve to the left. The hybrid case,
Fig.13A, displays a downward shift to the left. The GABA-only mechanism, Fig.13C, gives rise
to a nonlinear downward shift for background-on.
However, the downward shift is largest for
the right-most branch. This shift is inconsistent with the experimental result. Finally, for the
control case, Fig. 13D, where neither mechanism is present, there is no shift in the activation
curve when the background is illuminated.
It is instructive to point out that the control case can be deduced by solving for ICa in Eq.
(17) after setting ∂ICa/∂t = 0 (steady-state) with  = 0 = kG. The resulting expression
28
I Ca 
(V  E ) gCa
,
(1  e
) (1  kOCa [G])
C
Ca
 (VC  A)/ B
(23)
depends only on the state variable VC, since setting kG = 0 fixes the concentration of GABA
([G]) and makes [G] independent of changes in UH. Therefore, changes in the steady-state value
of UH, caused by background illumination, has no effect on Eq.(23) and this explains why there
is no shift whatsoever in the ICa -VC curve in Fig.13D.
In Fig. 11a of the experiment, the right-most branch of the curve corresponding to
background illumination is slightly above (nearly overlays) the background-off curve, and the
overall downward shift can be viewed as negligible. In our simulations, this would most closely
resemble the ephaptic-only case, or a weak hybrid case that favors an ephaptic mechanism (kG
small but nonzero). Our simulations predict that a GABA mechanism would have only a
negligible influence on the feedback mechanism. The experimental results are consistent with
our simulations. In the experiments, measurements were carried out in the presence of a high
concentration of picrotoxin, which is a GABA A receptor agonist that blocks GABA A – receptor
– mediated feedback responses (Verweij et al. 1996).
Without GABA, the background-
induced shift persisted.
In both the hybrid and ephaptic-only cases, Figs. 13 A and B, the steep part of the left branch
shows an increase in inward calcium current (ICa becomes more negative) when the background
is turned on. In our model, when the background is turned on, the cone potential within the
disk or center region does not change. The reason ICa changes is due to the spread of potential
in the horizontal cell network, from the surround to the center region. That is, the feedback
mechanism in the center region, driven by the horizontal cell potential, is responsible for the
change in ICa.
29
The above observation is related to the classic center-surround antagonism experiments by
Werblin and Dowling (1969).
In these experiments, first the center is turned on, then during
center stimulation the surround is turned on. In Fig. 14, a center spot is illuminated (t = 1000ms)
for the hybrid case corresponding to Fig.13A. The horizontal cell hyperpolarizes as seen in
Fig.14a. During center-on an annulus stimulus (surround) is then turned on (t = 3000ms).
Figure 14A shows that the horizontal cell potential at the recording site (middle of the center
region) further hyperpolarizes, this time due to the spread of potential through the horizontal cell
network from the annulus region to the center region.
In Fig.14B, at center-on, the inward
calcium current decreases (ICa less negative), but when the annulus is turned on the inward
calcium current increases (ICa more negative). This center-surround antagonism of ICa directly
affects glutamate. Therefore, [GL] in Fig.14C also exhibits center-surround antagonism.
30
DISCUSSION
We have formulated a partial differential equation system (Eqs. (14)-(21)) that includes the
horizontal cell syncytium and its numerous processes within cone pedicles, and have compared
simulation results with experimental data for square, slit, and disk test regions. Our goal in this
paper was to test the validity of the GABA vs. ephaptic hypotheses by comparing simulation
with experimental data for cat retina. We found that for square and slit stimulus regions an
ephaptic-only mechanism for calcium entry is consistent with the temporal dynamics associated
with background-induced flicker enhancement experiments in cat, whereas a GABA-only
mechanism is not consistent with experiments. However, we did find that GABA was useful for
modulating flicker enhancement in concert with the ephaptic mechanism. In fact, this hybrid
mechanism incorporating both ephaptic and GABA kinetics best fit the experimental data.
Figure 15 summarizes the hybrid mechanism’s simulated flicker-enhancement response for
several of the state variables (square-stimulus only).
To model the horizontal cell syncytium and its processes, we generalized a continuum spine
formulation originally developed by Baer and Rinzel (1991) for dendritic cables with many
spines. To justify the use of a continuum spine model for horizontal cells requires a sufficiently
large spine density. Indeed, for cat retina we used a spine density of n ≈ 3.2 × 104 spines per
electrotonic area.
We also determined from the literature an estimate for the spine stem
resistance of RSS  1300 M. A stem resistance this large suggests that the spine heads may be
electrically isolated from the slab.
Electrically isolated spines with excitable channels are
capable of generating action potentials. During action potential generation the potential in the
spine-heads is significantly different from the potential in the dendritic base, but during recovery,
31
or when the system is subthreshold, the potentials in the head and base are nearly equal (Segev
and Rall 1988; Baer and Rinzel 1991). Although the retina model is technically nonlinear, the
system behaves as though the membrane is passive and that is why in Fig. 15 the electrical
potential in the spine heads UH closely approximate the potential in the slab VH. The relative
error between the potentials can be deduced analytically from a simple reduction of the model
equations. Consider Eq. (14) only for the passive membrane case (Iion = VH / Rm). The uniform
steady-state of Eq. (14) reduces to RS n (UH - VH)/RSS = VH or in terms of the relative error
between the two potentials, ( UH - VH ) / VH = RSS / (RS n ). Recall that the horizontal cell gap
junction sheet resistance was RS  12M . Substituting the above estimates for RSS, RS, and n ,
the relative error is approximately 0.0034 or 0.34%. We found this value to be an accurate
relative error estimate between UH and VH in computations using the full nonlinear model.
In this paper we did not formulate a transfer function between light intensity and photocurrent
density. Instead we roughly estimated the photocurrent density based on how the simulations fit
the enhancement data. For example, in Fig.5 the amplitude of Iflick was chosen to be 7.15
A/cm2 (Aflick = -7.15), corresponding to a red 650 nm, 6.8 log quantam-2s-1 flickering square
test stimuli.
Likewise, we chose the magnitude of Ibkgd to be 7 A/cm2 (Abkgd
= -7)
corresponding to the diffuse background stimulus of wavelength 423 nm and irradiance 3.4 log
quantam-2s-1. In the experiment that plotted enhancement vs. flicker frequency, the flicker
irradiance was increased to 7.30 log quantam-2s-1 and the diffuse background irradiance was
increased to 4.40 log quantam-2s-1. In Fig.7 we compensated for this increase in both the
flicker and background irradiances by increasing the amplitude of Iflick to 7.7 A/cm2 and the
magnitude of Ibkgd to 7.4 A/cm2 . Similar adjustments were made for the phase shift study.
For slit test-regions, the only available experimental data was for the enhancement vs. flicker
32
frequency plot in Fig.4 of Pflug et al. (1990). In the experiment, the slit wavelength and
irradiance values were the same as the square test-region values in Fig. 1 of the same paper.
Therefore, for all slit simulations, we chose the amplitude of Iflick as 7.15 A/cm2 and the
magnitude of Ibkgd as 7 A/cm2 .
We found that an ephaptic-only mechanism was necessary for capturing major temporal
dynamics of background-induced flicker enhancement. That is, when we removed the ephaptic
mechanism by setting α = 0 in Eq. (17) for calcium activation, we could not reproduce the
temporal dynamics observed in flicker-enhancement experiments. Setting α = 0 effectively
decoupled the horizontal cell feedback network from the cone. In our formulation Ibkgd enters
the cone directly rather than, for example, entering the cone as a surround or background current
coming in through the horizontal cell network ( Ibkgd added to Eq. (14) instead of Eq. (21)).
This leads to a subtle but surprising result. We have discovered that biasing the cone synaptic
voltage directly with surround current can trigger enhancement provided a horizontal cell
feedback network is attached to the cone.
An equation for the rods would be necessary if the focus of this study was on the spatial
properties of enhancement.
The background current would then be moved from the cone
equation to a new equation for the rods. Additional terms for the rod-cone gap junction would
then be added to both the rod and cone equations. Furthermore, if hyperpolarized rods act
globally to shorten the horizontal-cell length constant by uncoupling gap junctions, an additional
equation for a time and space varying gap junction resistance may be needed as well.
Although the focus of this study was on the temporal properties of enhancement it was
necessary to reflect changes in stimulus geometry, size and flicker irradiance in order to compare
33
simulations with experiments. To compensate for these changes without over-complicating the
model we introduced the desensitization factor γ. Recall that γ was defined as a lumped
parameter that reduced the background current inside the stimulus region to account for both the
saturation desensitization of rods when exposed to bright flicker and to a much lesser extent
changes in the length constant due to background illumination. In a more complete model for
the spatial properties, as described above, γ would be a computed value rather than a model
parameter.
We found that changes in γ were consistent with changes in stimulus geometry and flicker
irradiance. When γ was tuned to fit the experimental data we assumed (a) a smaller stimulus
region implies a smaller desensitization, which in turn implies a larger value of γ; (b) a larger
flicker irradiance implies a smaller value of γ; (c) rod desensitization in a slit test region is
larger than that in a square test region of the same width and flicker irradiance. The simulations
in Fig.5 for background-induced flicker enhancement correspond to the experimental
configuration of a 250-µm square test-region with a flicker irradiance of 6.8 log quanta ˖ µm-2 ˖
s-1 (Fig.1 in Pflug et al. 1990). Here we tuned γ = 0.2277 for a reasonable fit to the data. In
Fig.7, for percent enhancement versus frequency, the stimulus size was reduced to a 250-µm
square test-region but the flicker irradiance was increased to 7.3 log quanta ˖ µm-2 ˖ s-1 (Fig.3 in
Pflug et al. 1990). Clearly, under assumptions (a) and (b), a smaller stimulus region and larger
flicker irradiance are opposing effects: either γ increases, decreases, or remains relatively
unchanged. Here, a reasonable fit to the data required an increase in the desensitization factor to
γ = 0.2492.
irradiance.
Therefore, for this case, a smaller stimulus region dominates a larger flicker
Finally, when changing from a 250-µm square test-region to a 250-µm slit test-
34
region, without changing the flicker irradiance, required a value of γ = 0.06 to obtain a
reasonable fit to the data (Fig.1 in Pflug et al. 1990). This change is consistent with assumption
(c) above.
In this paper we took a phenomenological approach to model the ephaptic mechanism. Recall
that the term UH (UH < 0) was inserted into Eq.(17) to translate the steady-state ICa vs. VC
curve to the left when the horizontal cell membrane potential is hyperpolarized. Recently
Gardner et al. (2013) tested the viability of the ephaptic hypothesis for a single synapse by means
of detailed numerical simulations. They applied a drift-diffusion model including membrane
boundary current equations to a two-dimension cross section of the triad synapse to demonstrate
that the feedback can be strictly electrical. They found that the strength of the feedback response
depended on the geometric configuration of the post synaptic processes within the triad synapse.
The results of the drift-diffusion simulations suggest a way of modifying our model to
represent the ephaptic mechanism more realistically. In the drift-diffusion simulations there
were three distinct isopotential regions: the horizontal cell spine stem and head, and the
intersynaptic space. Also, the intersynaptic space was isolated from the extracellular space by a
high resistance pathway between the cone (or bipolar cell) and the horizontal cell. This high
resistance pathway is analogous to the high resistance pathway produced by a spine neck. This
suggests, for our model, to remove the
UH
term in Eq.(17) and add a new equation
representing the intersynaptic space as an isopotential compartment separated from ground by a
coupling resistance. Although the details need to be worked out, this new approach shows
promise as a way to make the ephaptic component of our model less phenomenological.
35
Our simulations in Fig.5 demonstrated that our model captures ‘sag’ and post-inhibitory
rebound dynamics similar to those observed in flicker enhancement experiments. In further
testing of our model we found that when Iion and its auxiliary equations are removed the
enhancement persists and the magnitude remains unchanged, indicating that ‘sag’ and postinhibitory rebound, in our model, can be unrelated to feedback. Some studies use the ‘sag’ or
‘rollback’ response of horizontal cells as an index of feedback (Kamermans et al. 2001; Vessey
et al. 2005; Fahrenfort et al. 2009), whereas others contend that the ‘sag’ response is not a
sensitive measure of feedback (Klaassen et al. 2011). Without adding any other new mechanisms
to our model but removing Iion, we found that slowing down the GABA kinetics by making
kG << 1 can cause ‘sag.’ However, we have ruled out this specific mechanism for ‘sag’ since it
requires slowing down the GABA kinetics to speeds that are physiologically unrealistic.
Whether or not ‘sag’ dynamics is related to feedback requires further study.
Finally, we view our multiscale-continuum approach as a first step in formulating a multi-layer
mathematical model (partial differential equation system) of retinal circuitry. In this study we
have focused on the outer plexiform layer, but this model could be extended to include the other
‘brain nuclei’ within the retina: the inner plexiform layer where bipolar, amacrine,
interplexiform, and ganglion cells interact. Modeling the inner plexiform layer will present new
challenges. It is thicker than the outer plexiform layer, it has more synaptic contacts per unit
area, and possesses a greater variety of chemical synapses and gap junctions.
36
APPENDIX
In this APPENDIX we take advantage of the symmetry of slit, square, and disk stimuli to
prepare Eqs.(14)-(21) for numerical and analytic study. For each stimulus geometry we
formulate appropriate boundary conditions, explicitly define the functions that generate Iflick and
Ibkgd, and
specify the finite domain and boundary conditions for numerically integrating the
system.
Slit Stimulus
Nelson (1977) employed slits of light of variable width and position to characterize the
receptive fields of horizontal cells. The slit stimulus is convenient from an experimental and
theoretical point of view. Nelson recognized that if the slit is long enough, current flow in the
horizontal cell network is limited to one spatial dimension, namely perpendicular to the slit. To
model the slit stimulus it is convenient to use rectangular coordinates in the xy-plane. In
rectangular coordinates, ∇2VH = ∂2VH/∂x2 + ∂2VH/∂y2 in Eq.(14). An approximation that
reduces the computational complexity of the problem is to assume that the flicker stimulus is
applied over an infinite vertical strip (slit) of half width a. Consequently, VH and the other
dependent variables are constant in the y-direction (assuming all parameters are constant and
Ibkgd is constant in the y-direction). Therefore, ∂VH/∂y = 0 which implies that ∂2VH/∂y2 = 0, and
Eq.(14) is replaced by
2
VH
U  VH
2  VH
m

 RS n H
 I ion (VH , hV ) Rm .
2
t
x
RSS
(A1)
The other equations, (15)-(21), remain of the same form only now the state variables depend on x
and t rather than on x, y, and t. This reduction to one spatial dimension yields the same solution
as the two-dimensional problem with the advantage of saving considerable computation time
37
when solved numerically. Also note that Eq.(A1) has the same form as Baer and Rinzel’s
equation for a dendritic cable studded with a continuum of spines (Baer and Rinzel 1991),
allowing for some useful comparisons.
To numerically integrate the reduced model using a finite difference method, we require the x
domain to be finite; i.e.; -L ≤ x ≤ L, where L >> a. To reduce computation time, we take
advantage of the symmetry about x = 0 and solve the problem over the interval 0 ≤ x ≤ L. We
impose the following sealed end (homogeneous Neumann) boundary conditions:
VH
(0, t )  0
x
and
VH
( L, t )  0 .
x
(A2)
At x = 0, the sealed end is required to preserve the symmetry.
To generate the applied currents Iflick and Ibkgd, we first define an analytic approximation of the
unit step function, namely
H ( w,  ) 
1  tanh(  w)
,
2
(A3)
where β is a parameter that controls the slope of the transition between 0 and 1 about w = 0. The
function used to generate the “square” wave flicker stimulus in Eq.(21) is
flick
flick

 Aflick H ( z (t ), 1 ) H (a  x,  4 ) for tbeg  t  tend
I flick ( x, t )  
,
0
for t elsewhere


(A4)
flick
flick
where Aflick is the amplitude of Iflick in the hyperpolarizing direction, and tbeg
and tend
are the
beginning and end times of the flicker stimulus. The sine function,
 2
flick 
z (t )  sin 
(t  tbeg
) ,
 P

(A5)
when inserted for w in the step function H, generates a smooth approximation of a square wave
flicker stimulus of period P, as depicted in Fig. 3B. Parameter β1 in Eq.(A4) controls the slope
38
of the onset and offset of each flicker, and β4 controls the spatial slope of transition at the
boundary of the stimulus region of half width a. The function used to generate the background
stimulus is
bkgd
bkgd
Ibkgd ( x, t )  Abkgd H (t  tbeg
, 2 ) H (tend
 t , 3 ) [  (1   ) H ( x  a, 4 )] ,
(A6)
where Abkgd is the hyperpolarizing amplitude of Ibkgd exterior to the stimulus region (x ≥ a),
bkgd
bkgd
tbeg
and tend
are the beginning and end times of the background stimulus, β2 and β3 control the
slope of the onset and offset of the background stimulus, and γ is the desensitization factor
defined as the fraction to which Ibkgd is reduced within the stimulus region due to desensitization
of rods when exposed to bright flicker. The last factor in Eq.(A6), [γ + (1 – γ) H(x-a, β4)],
behaves as a smooth switch between γ for 0 < x < a, and 1 for x > a.
Square Stimulus
For a square stimulus region it is convenient to again use rectangular coordinates, but for this
geometry we cannot reduce the model to one spatial dimension. Therefore, all state variables in
Eqs.(14)-(21) are dependent on x, y and t, and in Eq.(14) we set ∇2VH = ∂2VH/∂x2 + ∂2VH/∂y2 .
Here, Iflick is applied over a region centered at the origin and bounded by the sides x = ±a and
y = ±a.
To numerically integrate the system using a finite difference method requires that the spatial
domain is finite. We choose this domain to be a square region bounded by x = ±L and
y = ±L
where L >> a. Just as for the slit stimulus case, we reduce the computation time by taking
advantage of the symmetry about the x and y axes and solving the problem over the domain
39
bounded by x = 0, x = L, and y = 0, y = L. We impose homogeneous Neumann boundary
conditions given by
VH
VH
(0, y, t )  0 and
( L, y, t )  0 for 0  y  L
x
x
(A7)
VH
VH
( x, 0, t )  0 and
( x, L, t )  0 for 0  x  L .
y
y
(A8)
and
The function defining the flicker stimulus over a square region is a two-dimensional
extension of Eq.(A4) given by
flick
flick

 A H ( z (t ), 1 ) H (a  x,  4 ) H (a  y,  4 ) for tbeg  t  tend
I flick ( x, y, t )   flick
,
0
for
t
elsewhere


(A9) where H is defined in Eq.(A3) and z in Eq.(A4). The function used to generate the
background stimulus is
bkgd
bkgd
Ibkgd ( x, y, t )  Abkgd H (t  tbeg
, 2 ) H (tend
 t , 3 )[  (1   ) F2 ( H ( x  a, 4 ), H ( y  a, 4 ))],
(A10)
where F2 is defined by
F2 ( p, q) 
| pq|| pq|
.
2
(A11)
When p = H(x − a, β4) and q = H(y − a, β4), F2 behaves as a two-dimensional smooth unit
step function where F2 = 0 inside the square flicker region and F2 = 1 outside the region.
Disk Stimulus
The disk stimulus is a circular region of radius a centered at the origin. Like the slit
stimulus, the model equations for the disk can be reduced to one spatial dimension. Here the
40
reduction to one dimension is appropriate if we assume that all the state variables and the initial
conditions are circularly symmetric. That is, the variables are constant on each circle concentric
with the disk, so that the solution to Eqs.(14)-(21) is fully represented along the radii r of the
disk. For this circularly symmetric case ∇2VH = ∂2VH/∂r2 + (1/r) ∂VH/∂r, where r = (x2 + y2)1/2,
and Eq.(14) becomes
m
VH
 2V
U  VH
1 VH
  2 ( 2H 
)  RS n H
 I ion (VH , hV ) Rm ,
t
r
r r
RSS
0  r  .
(A12)
The other equations, (15)-(21), remain of the same form only now the state variables depend on
r and t rather than on x, y and t. The boundary condition at the origin requires that VH is
bounded as r → 0+ . Likewise, VH is required to be bounded as r → ∞.
In order to implement the finite difference method, we impose the finite domain 0 < r < R
where R >> a. At r = R, we choose a zero-flux boundary condition in the radial direction.
Although Eq.(A12) is singular at r = 0, a twice continuously differentiable solution does exist
over the entire domain, including the origin. Therefore, to numerically approximate VH at the
origin, it is necessary to discretize r over the grid points ri using half-steps. That is, we set
ri = h/2 + ( i – 1) h , where h is the distance between grid points. As a consequence of the
circular symmetry, the centered difference approximation for ∂VH/∂r is zero at the origin.
A ghost (or fictional) half-step approximation of VH, in the negative r direction, is introduced
for the centered difference approximation at r = 0.
The flicker and background currents have the same functional forms as Eqs. (A4) and (A6),
with r replacing x. One simulation of interest in this paper is the feedback-induced shift in
calcium activation. Calcium current-voltage relationships are computed with a steady disk-
41
shaped center stimulus region in the presence and absence of a full field background
illumination. Here the flicker stimulus Iflick is replaced by the steady stimulus Istdy defined by

 A H (a  r ,  4 )
I stdy (r , t )   stdy
0


stdy
stdy
for tbeg
 t  tend
.
(A13)
for t elsewhere
The background stimulus Ibkgd remains of the same form as Eq.(A6) with r replacing x.
GRANTS
This work was supported in part by the National Science Foundation under grant DMS-0718308.
DISCLOSURES
No conflict of interest, financial or otherwise, are declared by the authors.
42
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51
FIG. 1. The dynamics of background-induced flicker enhancement. The left panels show
simulated enhancement dynamics for square test-regions, the right panels for slit test-regions.
A: The enhancement effect. The thin bar on the time axis denotes the duration of the flicker
stimulus, the thick bar denotes the duration of the background stimulus. For both square and slit
test-regions, the horizontal cell flicker response amplitude increases during background
illumination. In addition, while the background is on, there is a slow repolarization or ‘sag’
effect, followed by an overshoot. Compare with Fig.1 in Pflug et al. (1990) for the experimental
results. B: The “two-limb” feature of the percent enhancement versus frequency curves, for
experiment (dotted) and simulation (solid). Data points for experiments obtained from Pflug et
al. (1990). C: Background-induced phase shift in flicker waveform between a flicker cycle in
the dark (solid) and a flicker cycle with background illuminated (dashed). Compare to Fig. 5 in
Pflug et al. (1990).
FIG. 2. Estimate of spine density and spine stem resistance for horizontal cell syncytium.
A: Spiny processes emanate from the horizontal cell syncytium (stippled region), to form
synapses with cones (not shown). The variables UH and VH denote the horizontal cell membrane
potentials in the spine head and syncytium respectively. The spine density is estimated to be
n ≈ 3.2 × 104 spines per electrotonic area, sufficiently large to justify a continuum spine
formulation. B: Using the simplifying assumption that the spine stem is cylindrical with stem
length Lss ≈ 5μm and stem diameter dss ≈ 0.1 μm, the stem resistance is estimated to be
Rss ≈ 1300 MΩ. The electrical isolation caused by this large stem resistance is offset by the large
spine density. Consequently, at a given location (x,y), UH and VH are approximately equal.
See text for details of the above estimates.
52
FIG. 3. The functions used to define both Iflick and Ibkgd depend on the geometry of the
stimulus region. The explicit forms of these functions, for slit, square, and disk test-regions, are
given in the APPENDIX.
Here we show, as an example, the time course and spatial
dependency of the flicker stimulus and background illumination for a slit test-region. A: The
flicker stimulus is applied over an infinite vertical slit (stippled) of half width a. B: The square
wave flicker stimulus is plotted as a function of time. The minimum of each square wave is
denoted by Aflick. We model the onset and offset of each wave cycle as a smooth sigmoidal
transition, as shown in the magnified insets. C: The minimum of the square wave flicker
stimulus as a function of x. Here, the insets magnify the smooth spatial transition at the
boundaries of the stimulus region x = ±a. D: The background current Ibkgd is displayed as a
function of time for both inside and outside the stimulus region S. The parameter Abkgd is
defined as the minimum of Ibkgd outside of S. Inside S, we account for both the saturation
desensitization of rods when exposed to bright flicker and changes in the length constant due to
background illumination by changing Abkgd to γ Abkgd, where 0 ≤ γ ≤ 1. Note that the offset
of Ibkgd is set slower than its onset to account for rod after-effect.
53
FIG. 4. Both ephaptic and GABA feedback mechanisms are modeled by Eqs. (14) - (21).
Arrows numbered (1)-(8) summarize the dynamics underlying the background-induced flicker
enhancement experiments. (1) Both flicker stimulus Iflick and background stimulus Ibkgd drive
the cone membrane potential VC (Eq. (21)). The flicker stimulus, transduced by the cone,
directly modifies the cone membrane potential, whereas the background stimulus, transduced by
the rods, indirectly modifies the cone membrane potential via rod-cone gap junctions. In the
background-induced flicker enhancement experiments, first the flicker stimulus is turned on until
the system reaches steady-state oscillations. Next, the background stimulus is applied. (2) The
background illumination further hyperpolarizes the cone resulting in a change in calcium
activation (Eq.(17)). (3) When calcium current enters the cone ( ICa < 0 ) there is an increase in
the release of transmitter glutamate [GL] into the synaptic cleft (Eq. (16)). (4) Glutamate
depolarizes the already negative horizontal cell spine head membrane potential UH (Eq. (15)).
(5) The spread of potential from other slab locations can also change VH and therefore UH (Eqs.
(14) and (15)). (6) Under the ephaptic hypothesis, a depolarizing UH reduces the influx of ICa
into the cone (Eq. (17)) and, via the feedback loop (6) → (3) → (4) → (6), drives UH in the
opposite, hyperpolarizing direction.
(7) Under the GABA hypothesis, a depolarizing UH
increases the release of calcium blocker GABA (Eq. (18)), reducing the influx of ICa into the
cone. Here, the feedback loop (7) → (8) → (3) → (4) → (7) drives UH in the opposite,
hyperpolarizing direction.
54
FIG.5. Background-induced flicker enhancement (Square test-region). Each panel shows
the simulated horizontal cell response (VH) to a flickering 16 Hz, 250-m square test-region
stimulus (duration denoted by thin bar) and a superimposed steady background stimulus
(duration denoted by thick bar). The strength of the feedback mechanisms is controlled by in
Eq. (17) and kG in Eq. (18). Here, the desensitization factor is γ = 0.2277; other parameters
are listed in Tables 1-3. Compare each panel to the experimental microelectrode record in Fig. 1
of Pflug et al. 1990. A: A hybrid of both GABA and ephaptic kinetics ( kG = 1, )
induces an enhancement of E = 99. B: For ephaptic-only kinetics (kG = 0,  the
enhancement is reduced to E = 87. C: The response for GABA-only kinetics (kG = 1 ,
ere, the enhancement decreases significantly to E = 22. D: When neither mechanism
is present (kG = 0, the flicker response amplitude during background-on is less than that
when the background is off, yielding a slightly negative enhancement of E = -0.75. This
control case demonstrates that a feedback mechanism is necessary to significantly amplify the
enhancement, and both ‘sag’ and post-inhibitory rebound dynamics are present when the
feedback mechanisms are blocked.
55
FIG. 6. Background-induced flicker enhancement (Slit test-region). Each panel shows the
simulated horizontal cell response (VH) to a flickering 16Hz, 250m slit test-region stimulus
(duration denoted by thin bar) and a superimposed steady background stimulus (duration denoted
by thick bar). Here the desensitization factor for the slit geometry is γ = 0.06. See Fig. 5 for
other parameter values. A: The hybrid mechanism ( kG = 1, ) exhibits an enhancement
of E = 37. This is the same case shown in Fig.1A right. B: The ephaptic-only case ( kG = 0,
) enhances the flicker to E = 32. C: The GABA-only case ( kG = 1, ) enhances the
flicker to E = 35. D: When neither mechanism is present ( kG = 0, ), the enhancement drops
to 23. The dynamics for all four cases are consistent with experimental microelectrode records
for slit-stimuli described in the literature (Nelson et al. 1990).
FIG. 7. Percent enhancement as a function of frequency (Square test-region). Each panel
compares simulated percent enhancement (solid curves) with experiment (dotted curve) for a
150-m square test-region at different flicker frequencies. Here,  = 0.2492, Aflick = -7.7A/cm2,
and Abkgd = -7.4A/cm2. See Fig. 5 for other parameter values. This figure demonstrates that the
hybrid and ephaptic-only mechanisms best fit the experimental data in Fig. 4 of Pflug et al.
(1990). A: Hybrid ( kG = 1, ): This is the same case shown in Fig.1B left. B: Ephapticonly ( kG = 0, ).
( kG = 0, ).
C: GABA-only ( kG = 1, ).
D: Neither mechanism
56
FIG. 8. Percent enhancement as a function of frequency (Slit test-region). Consistent with
Fig. 7 for the square-test region, the hybrid and ephaptic-only kinetics for the slit test-region
provide the best fit to the experimental data in Fig.4 of Pflug et al. (1990). See Fig.6 for other
parameter values. A: Hybrid ( kG = 1, ): this is the same case shown in Fig.1B right; B:
Ephaptic-only ( kG = 0, ). C: GABA-only ( kG = 1, ). D: Neither mechanism
( kG = 0, ).
FIG. 9.
Flicker response amplitude decreases monotonically with increasing frequency.
Simulations confirm the experimental result (Fig. 3 in Pflug et al. 1990) that although
enhancement increases with frequency, the flicker response amplitudes both before and during
the background presentation decrease monotonically. Same parameters as Fig. 7A: 150-m
square test-region with a hybrid of both GABA and ephaptic kinetics.
FIG. 10. Simulation of background-induced phase shifts (Square test-region). A flicker
cycle in the dark (solid curve) is superimposed over a flicker cycle in the presence of steady
background illumination (dashed curve). To facilitate the comparison, the peak-to-peak dark
flicker response amplitude has been scaled to the peak-to-peak amplitude for background-on.
Here, the horizontal cell is responding to a 17.39Hz, 424-m square test stimulus, with  =
0.1915, Aflick = -7.7cm2, Abkgd = -7.25cm2. The remaining parameters are the same as in
Fig.5. The A: hybrid (same as Fig.1C left) and the B: ephaptic-only case display phase shifts to
the left, which are consistent with the experimental result (compare to Fig.5 in Pflug et al. 1990).
In sharp contrast, phase shifts are to the right in C: GABA-only and in D: neither mechanism
present.
57
FIG. 11. Simulation of background-induced phase shifts (Slit test-region). As in Fig.10, a
flicker cycle in the dark (solid curve) is superimposed over a flicker cycle in the presence of a
steady background (dashed curve). To facilitate the comparison, the peak-to-peak dark flicker
response amplitude has been scaled to the peak-to-peak amplitude for background-on. For the slit
test-region, the horizontal cell is responding to a 16Hz, 250m slit test stimulus. The remaining
parameters are the same as in Fig.6. Consistent with Fig.10 for square test-regions,
the A:
hybrid (same as Fig.1C right) and B: ephaptic-only cases display phase shifts to the left, which
are consistent with the experimental result for square test-regions (compare to Fig.5 in Pflug et
al. 1990). In sharp contrast, phase shifts are to the right in C: GABA-only and in D: neither
mechanism present.
58
FIG.12. Background-induced phase shifts depend on the rate of glutamate reuptake.
Parameter values are the same as in Fig.10 (square test-region). Here the background-induced
phase shifts of ICa , [GL] , and UH are compared for a ten-fold increase in the glutamate
reuptake rate. As in Fig.10, a flicker cycle in the dark (solid curve) is superimposed over a
flicker cycle in the presence of steady background illumination (dashed curve), and peak-topeak dark flicker response amplitude has been scaled to peak-to-peak amplitude for backgroundon. A: When neither mechanism is present (α = 0, kG = 0), increasing the glutamate reuptake
rate by a factor of ten ( by dividing kCa = 15 and τGL = 18.18 in Eq. (16) by ten ) has no effect
on the phase shift dynamics of inward calcium current ICa (left panels) . However, glutamate
(middle panels) does show a change in phase shift dynamics corresponding to an overall
decrease in concentration that is particularly pronounced during synaptic release (t > 35). The
shift in [GL] causes a modest shift in phase of the UH trajectory (right panels). B: For ephapticonly kinetics (α =1, kG = 0) there is a significant change in the shift dynamics of ICa when
glutamate reuptake is increased.
It is most evident when the inward calcium current is
decreasing (t < 30 or t > 40). The shift in ICa contributes to the large shift observed in both
[GL] and UH : a shift larger than that observed when neither mechanism is present.
59
FIG. 13. Background-induced shift in calcium current (Non-flickering disk). Each panel
shows the simulated steady-state dependence of cone calcium current ICa as a function of cone
potential VC at the center of a constant (non-flickering) circular disk (spot) stimulus. The
diameter of the disk is 65m, γ = 0.2676, and ICa represents the average current for a single
synapse. Other parameter values are listed in Tables 1-3. The plot of ICa vs. VC
for a spot
stimulus with no background (solid curve) is superimposed over the curve for a spot stimulus
with background illumination (dashed curve). A: Hybrid mechanism: Background illumination
shifts the ICa vs. VC curve downward over the range
-80 < VC < 0, allowing more calcium
current to enter the cone. This feedback-induced shift is nonlinear in the sense that the shift is not
purely a translation to the left of the original background-off ICa vs. VC curve. B: Ephapticonly: Here the shift is essentially a pure translation to the left of the background-off ICa vs. VC
curve. Consequently, the inward calcium current is slightly decreased for -56 < VC < 0 but
increased for -80 < VC < -56. C: GABA-only: The range of increased inward current is
reduced to -60 < VC < 0 compared to that range in the hybrid case (A above). Unlike the
ephaptic-only case there is no decrease in calcium current, but rather a pronounced increase in
inward calcium current over the range -40 < VC < 0. D: Neither mechanism: In the simulation
both the ephaptic and GABA mechanisms are blocked, resulting in no shift in calcium current.
60
FIG. 14. Both ICa and [GL] exhibit center-surround antagonism. For the hybrid case in
Fig. 13A, a center (spot) stimulus of radius 50µm is applied for 1000 < t < 7000 ms. During
center-on, an annulus stimulus of inner radius 60µm and outer radius 260 µm is applied for
3000 < t < 5000 ms.
The magnitudes of both the center and the annulus stimuli are
Astdy = -7 µm/cm2, otherwise Astdy = 0. Other parameter values are the same as in Fig.13A.
A: At center-on (t = 1000 ms), the horizontal cell hyperpolarizes to VH  -29.18 mV. When the
annulus is turned on (t = 3000 ms) the horizontal cell hyperpolarizes further; sharply down to
VH -32.40 mV. B: The calcium current ICa exhibits center-surround antagonism. At centeron, the calcium current changes from
ICa  -1.34 to -0.26 pA; an approximate 1.08 pA
decrease in the inward current. However, when the annulus is turned on, the inward calcium
current reverses direction and increases. C: Consequently, glutamate concentration [GL] also
exhibits center-surround antagonism: the decrease in inward calcium current at center-on drives
the glutamate concentration down to [GL]  3.77 µM,
but at annulus-on the glutamate
concentration reverses and increases to [GL]  8.29 µM.
FIG. 15. Simulation of background induced flicker enhancement (Hybrid mechanism) for
state variables VC , UH , ICa , VH , [GL]
and [G].
Parameter values are the same as in
Fig. 5A. The panels on the right demonstrate that the potential VH in the horizontal cell slab
closely approximates the potential UH in the spine head, and that the time course for [G] is
similar to both UH and VH . However, the time course of glutamate [GL] and the calcium
current ICa are more similar to the time course of the cone potential VC .
61
Square test-region
A
Slit test-region
-25
VH (mV)
VH (mV)
-25
-30
-35
-35
1000 2000 3000 4000 5000
t (ms)
experiment
simulation
200
100
VH (mV)
0
5
experiment
simulation
200
100
10
20 30 40
Frequency (Hz)
-28
0
5
10
20 30 40
Frequency (Hz)
-28
in the dark
with bkgd
-30
-32
-34
1000 2000 3000 4000 5000
t (ms)
300
0
10
20
30 40
t (ms)
VH (mV)
Percent
Enhancement
300
0
Percent
Enhancement
0
B
C
-30
-30
-32
-34
50
Figure 1:
in the dark
with bkgd
0
10
20
30 40
t (ms)
50
60
62
Horizontal cell syncytium and spine processes
UH
x
A
VH
dss
B
Lss
0
Figure 2:
63
y
A
B
S
-a
x
a
0
Iflick (µA/cm2)
0
Stimulus
Region
Aflick
0
0
Aflick
Aflick
961
800
C
t
965
900
992
t (ms)
t
996
1000
1100
D
(µA/cm )
2
2
min Iflick (µA/cm )
0
0
Aflick
inside S
γ Abkgd
0
Ibkgd
Aflick
0
Aflick
Abkgd
outside S
a
-a
-a
a
1500
x (µm)
2000
2500
3000
t (ms)
Figure 3:
3500
4000
64
Model variables with summary of feedback mechanisms
(1)
VC cone potential (mV)
Iflick (mA/cm2)
Ibkgd (mA/cm2)
(2)
ICa calcium current (pA)
(3)
Ephaptic
(6)
feedback
(8) GABA
[GL] glutamate (mM)
feedback
(4)
UH membrane potential of
HC spine head (mV)
(5)
VH membrane potential of
HC slab (mV)
Figure 4:
(7)
[G] GABA (mM)
65
Background-Induced Flicker Enhancement (Square test-region)
E ≈ 99
-40
0
VH (mV)
-30
-40
E ≈ 22
0
Ephaptic only
-20
-30
E ≈ 87
-40
-50
1000 2000 3000 4000 5000
GABA only
C -20
-50
VH (mV)
-30
-50
B
Hybrid mechanism
-20
0
-30
-40
-50
1000 2000 3000 4000 5000
t (ms)
1000 2000 3000 4000 5000
Neither mechanism
D -20
VH (mV)
VH (mV)
A
E ≈ -0.75
0
1000 2000 3000 4000 5000
t (ms)
Figure 5:
66
Background-Induced Flicker Enhancement (Slit test-region)
E ≈ 37
-40
0
VH (mV)
-30
-40
E ≈ 35
0
Ephaptic only
-20
-30
E ≈ 32
-40
-50
1000 2000 3000 4000 5000
GABA only
C -20
-50
VH (mV)
-30
-50
B
Hybrid mechanism
-20
0
-30
-40
-50
1000 2000 3000 4000 5000
t (ms)
1000 2000 3000 4000 5000
Neither mechanism
D -20
VH (mV)
VH (mV)
A
E ≈ 23
0
1000 2000 3000 4000 5000
t (ms)
Figure 6:
67
Percent Enhancement vs. Frequency (Square test-region)
Percent Enhancement
experiment
simulation
200
100
0
-100
C
Percent Enhancement
B
300
Hybrid
mechanism
5
10
20
30 40
200
100
0
-100
GABA
only
5
10
20
30 40
Frequency (Hz)
300
200
100
0
-100
D
300
Percent Enhancement
Percent Enhancement
A
Ephaptic
only
5
10
20
30 40
20
30 40
300
200
100
0
-100
Neither
mechanism
5
10
Frequency (Hz)
Figure 7:
68
Percent Enhancement vs. Frequency (Slit test-region)
Hybrid
mechanism
100
0
-100
5
10
20
30 40
200
GABA
100
only
0
5
10
20
30 40
Frequency (Hz)
300
200
Ephaptic
only
100
0
-100
D
300
-100
Percent Enhancement
experiment
simulation
200
C
Percent Enhancement
B
300
Percent Enhancement
Percent Enhancement
A
5
10
20
30 40
20
30 40
300
200
Neither
mechanism
100
0
-100
5
10
Frequency (Hz)
Figure 8:
69
Flicker response amplitude decreases as frequency increases
-20
E ≈ 60
10 Hz
E ≈ 115
20 Hz
E ≈ 175
30 Hz
E ≈ 290
40 Hz
-30
-40
-20
VH (mV)
-30
-40
-20
-30
-40
-20
-30
-40
0
1000
2000
3000
t (ms)
Figure 9:
4000
5000
70
Background-induced phase shift (Square test-region)
Hybrid mechanism
-28
Ephaptic only
-28
-30
VH (mV)
VH (mV)
in the dark
with bkgd
-32
-34
0
10
30
40
VH (mV)
-32
0
10
20
30 40
t (ms)
0
Figure 10:
20
30
40
50
-30
-32
-34
50
10
Neither mechanism
-28
-30
-34
-32
-34
50
GABA only
-28
VH (mV)
20
-30
0
10
20
30 40
t (ms)
50
71
Background-induced phase shift (Slit test-region)
Hybrid mechanism
-28
Ephaptic only
-28
-30
VH (mV)
VH (mV)
in the dark
with bkgd
-32
-34
0
10
20
30
40
50
-30
-32
-34
60
0
GABA only
20
30
40
50
60
Neither mechanism
-28
VH (mV)
-28
VH (mV)
10
-30
-32
-30
-32
-34
-34
0
10
20
30 40
t (ms)
50
60
Figure 11:
0
10
20
30 40
t (ms)
50
60
72
[GL] reuptake influences background-induced phase shifts
[GL] (µM)
-2
ICa (pA)
-35
-36
Ephaptic
only
Increased
rate of [GL]
reuptake
-34
[GL] (µM)
0 10 20 30 40 50
Increased
rate of [GL]
reuptake
-27
0
3
-2
0
-1
-34
0 10 20 30 40 50
30
-1
Neither
mechanism
0 10 20 30 40 50
[GL] (µM)
ICa (pA)
0
0 10 20 30 40 50
0
0
3
UH (mV)
-1
-2
B
[GL] (µM)
ICa (pA)
-2
0
-27
UH (mV)
ICa (pA)
-1
in the dark
with bkgd
UH (mV)
30
0
UH (mV)
A
0
0 10 20 30 40 50
t (ms)
Figure 12:
-35
-36
0 10 20 30 40 50
73
Background-induced shift in ICa (Non-flickering disk)
A
spot
spot + bkgd
0
-0.5
nonlinear
shift
-1
-1.5
-2
-100
-80
-60
-40
-20
D
-80
-60
-40
-20
0
Neither mechanism
0
ICa (pA)
ICa (pA)
0
nonlinear
shift
-1.5
-2
-100
pure
translation
-1
-2
-100
0
GABA only
-1
-0.5
-1.5
C
-0.5
Ephaptic only
0
ICa (pA)
ICa (pA)
B
Hybrid mechanism
-0.5
no shift
-1
-1.5
-80
-60
-40
-20
0
VC rest potential (mV)
-2
-100
-80
-60
-40
-20
VC rest potential (mV)
Figure 13:
0
74
Center-surround antagonism
A
VH (mV)
-25
center
on
center +
annulus
on
-29
annulus
off
-33
0
ICa (pA)
B
-0.5
-1
-1.5
-2
C
[GL] (µM)
24
16
8
0
1000
3000
5000
t (ms)
Figure 14:
7000
75
Background-Induced Flicker Enhancement (Hybrid mechanism)
-25
UH (mV)
VC (mV)
0
-50
-100
-35
0
1000 2000 3000 4000 5000
-1
1000 2000 3000 4000 5000
0
1000 2000 3000 4000 5000
0
1000 2000 3000 4000 5000
-30
-35
0
1000 2000 3000 4000 5000
25
2
[G] (µM)
[GL] (µM)
0
-25
VH (mV)
ICa (pA)
0
-2
-30
15
5
0
1000 2000 3000 4000 5000
t (ms)
1.6
1.2
t (ms)
Figure 15:
76
Table 1.
Reference parameter values for model equations
Symbol
Reference value
Ash
1.31 µm2
A
-40.8 mV
B
3 mV
Cm
1 µF/cm2
α
0.88
λ
250 µm
σh
-2.4 mV
τCa
5 ms
τGL
18.18 ms
τh
800 ms
τm
7.5 ms
τG
15 ms
θh
-30 mV
ECa
120 mV
ECl
-65 mV
Esag
120 mV
ELC
-68 mV
ELH
-60 mV
[Gi ]
5 µM
gCa
0.03 nS
gsag
1.7 × 104 nS/cm2
gLC
1.5 × 105 nS/cm2
gLH
105 nS/cm2
Idark
6.4 µA/cm2
kCa
15 µM/pA
kG
1 µM/mV
kOCa
1 µM−1
ksyn
1.572 × 10−13 (Ω · µM)−1
L
1280 µm
n
2.4 × 104
ni
1
T
310.15 K
Rm
7500 Ω · cm2
Rs
12 MΩ
Rss
1273.24 MΩ
Parameter
Surface area of horizontal cell spine head
Shift potential in sigmoidal activation curve
Scale potential in sigmoidal activation curve
Membrane capacitance
Dimensionless ephaptic feedback coefficient
Length constant of the horizontal cell sheet
Kinetic parameter in h equation for Iion
Time constant for calcium entering the cone
Time constant for glutamate release from cone
Time constant in h equation for Iion
Membrane time constant for horizontal cell membrane
Time constant for GABA release
Threshold of h∞ in h equation for Iion
Reversal potential for calcium current
Reversal potential for chloride current
Reversal potential for the “sag” current
Leak potential for cone
Leak potential for horizontal cell
Horizontal cell intracellular concentration of GABA
Peak calcium conductance
Conductance for the “sag” current
Leak conductance for cones
Leak conductance for horizontal cells
Dark current into cone
Scale factor for glutamate dependence on calcium
Scale factor for GABA release from horizontal cell
Binding equilibrium constant for GABA
Scale factor for synaptic glutamate current
Location of boundary condition for slit model
Electrotonic spine density (# of spines for λ2 sheet area)
Number of charges transported per GABA ion
Temperature
Passive membrane resistivity
Gap junctional sheet resistance
Spine stem resistance
77
Table 2.
Initial value of each state variable
Symbol
VH
UH
VC
[G]
ICa
[GL]
hV
hU
Initial value
-28.32 mV
-28.24 mV
-25.33 mV
1.629 µM
-1.375 pA
20.62 µM
0.3321
0.325
State variable
Horizontal cell slab membrane potential
Horizontal cell spine head membrane potential
Cone membrane potential
Concentration of GABA released by horizontal cells
Cone calcium current (ICa > 0 outward; ICa < 0 inward)
Concentration of glutamate released from the cone
Gating variable for the sag current in the horizontal cell slab
Gating variable for the sag current in the horizontal cell spine heads
78
Table 3.
Reference parameters for If lick , Ibkgd , and Istdy
Symbol
Af lick
Abkgd
Astdy
β1
β2
β3
β4
P
lick
tfbeg
lick
tfend
tbkgd
beg
tbkgd
end
tstdy
beg
tstdy
end
Reference value
-7.15 µA/cm2
-7 µA/cm2
-7 µA/cm2
50
0.15 ms−1
0.01 ms−1
0.28 µm−1
62.5 ms
900 ms
4564.29 ms
2121.43 ms
3342.86 ms
900 ms
11564.29 ms
Parameter
Factor controlling amplitude of If lick
Factor controlling magnitude of Ibkgd exterior to stimulus region S
Factor controlling magnitude of Istdy (for disk)
Dimensionless factor controlling onset & offset speed of each flicker
Factor controlling onset speed of Ibkgd
Factor controlling offset speed of Ibkgd
Factor controlling spatial slope of If lick & Ibkgd across S boundary
Period of “square” wave flicker stimulus
Beginning of flicker stimulus
End of flicker stimulus
Beginning of background illumination
End of background illumination
Beginning of steady disk stimulus
End of steady disk stimulus