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Physiological Reports. 2021;00:e14709.
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1 of 18
https://doi.org/10.14814/phy2.14709
wileyonlinelibrary.com/journal/phy2
1
|
INTRODUCTION
Midbrain dopamine (DA) neurotransmission is essential for
the control of voluntary movement (Sharples et al., 2014),
motivation (Bromberg-Martin et al., 2010) and is hypothe-
sized to be a key mediator of addictive behaviors (Kalivas
& Volkow, 2007; Wise, 2004). DA releasing neurons in-
volved in motor function are located in the substantia nigra
pars compacta and their axons mainly innervate the dorsal
striatum (Smith & Kieval, 2000), whereas those associated
with limbic and cognitive functions are found in the ventral
tegmental area (VTA), and primarily project to the nucleus
Received: 21 September 2020
|
Revised: 16 November 2020
|
Accepted: 14 December 2020
DOI: 10.14814/phy2.14709
ORIGINAL RESEARCH
Computational and theoretical insights into the homeostatic
response to the decreased cell size of midbrain dopamine neurons
FranciscoArencibia-Albite1,2
|
Carlos A.Jiménez-Rivera1
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original
work is properly cited.
© 2020 The Authors. Physiological Reports published by Wiley Periodicals LLC on behalf of The Physiological Society and the American Physiological Society
1Department of Physiology, University of
Puerto Rico, San Juan, Puerto Rico
2Department of Natural Sciences,
University of Sacred Heart, San Juan,
Puerto Rico
Correspondence
Francisco Arencibia-Albite, Department
of Natural Sciences, University of Sacred
Heart, San Juan 00914-8505, Puerto
Rico.
Email: franciscom.arencibia@sagrado.
edu
Funding information
National Institute of General Medical
Sciences, Grant/Award Number:
2SC1GM084854-05A1; National Science
Foundation, Grant/Award Number: OISE-
1545803
Abstract
Midbrain dopamine neurons communicate signals of reward anticipation and attri-
bution of salience. This capacity is distorted in heroin or cocaine abuse or in condi-
tions such as human mania. A shared characteristic among rodent models of these
behavioral disorders is that dopamine neurons in these animals acquired a small size
and manifest an augmented spontaneous and burst activity. The biophysical mecha-
nism underlying this increased excitation is currently unknown, but is believed to
primarily follow from a substantial drop in K+ conductance secondary to morphology
reduction. This work uses a dopamine neuron mathematical model to show, surpris-
ingly, that under size diminution a reduction in K+ conductance is an adaptation that
attempts to decrease cell excitability. The homeostatic response that preserves the
intrinsic activity is the conservation of the ion channel density for each conductance;
a result that is analytically demonstrated and challenges the experimentalist tendency
to reduce intrinsic excitation to K+ conductance expression level. Another unexpected
mechanism that buffers the raise in intrinsic activity is the presence of the ether-a-go-
go-related gen K+ channel since its activation is illustrated to increase with size re-
duction. Computational experiments finally demonstrate that size attenuation results
in the paradoxical enhancement of afferent-driven bursting as a reduced temporal
summation indexed correlates with improved depolarization. This work illustrates, on
the whole, that experimentation in the absence of mathematical models may lead to
the erroneous interpretation of the counterintuitive aspects of empirical data.
KEYWORDS
capacitance, cell size, computational modeling, dopamine neurons
2 of 18
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ARENCIBIA-ALBITE ANd JIMÉNEZ-RIVERA
accumbens in the ventral striatum and the prefrontal cortex
(Neuhoff et al., 2002). In the brain slice preparation, dopami-
nergic neurons express regular or irregular single-spike spon-
taneous activity in the range 1–7Hz (Grace & Onn, 1989;
Ping & Shepard, 1996). In freely moving (Hyland et al.,
2002) and anesthetized (Lee et al., 2004) rats, however, affer-
ent-driven bursts of action potentials are superimposed onto
this autonomous background. Such a burst signal produces
a transient increase in DA concentration greater than that
of intrinsically evoked single-spike activity (Chergui et al.,
1996; Wightman & Heien, 2006). In animal behavior stud-
ies the burst discharge is observed at the moment the animal
receives an unpredicted reward or is unexpectedly presented
with the opportunity to initiate an action that ends in reward
(Schultz, 1998, 2002). DA neurons are, therefore, thought to
convey signals of reward anticipation and attribution of sa-
lience (Schultz, 2006).
The capacity of DA neurons to encode and predict reward
may be substantially transformed during chronic exposure to
drugs of abuse or in conditions such as human mania. In mice
and rats repeated morphine, heroin or cocaine administration
decreases the size of VTA DA neurons (Arencibia-Albite
et al., 2012, 2017; Mazei-Robison et al., 2011; Russo et al.,
2006). In the case of morphine and heroin, the emergence of
a diminished morphology correlates with the development of
reward tolerance, which may help to explain the escalation in
drug use (Russo et al., 2006). Moreover, mice with a mutation
in the Clock gene have smaller VTA DA neurons and manifest
behavioral measures of mood, anxiety, activity, and reward
that are remarkably similar to bipolar patients in the manic
state (Coque et al., 2011; Mcclung et al., 2005; Roybal et al.,
2007). These genetically altered mice also show hyperactive
and hyperhedonic traits that are abolished as normal neuronal
dimensions are rescued with lithium treatment (Coque et al.,
2011). A unifying attribute among drug-treated mice and
Clock mutants is that their smaller VTA DA neurons have an
increased autonomous firing and bursting, in contrast, to sim-
ilar neurons from wild-type littermates (Coque et al., 2011;
Mazei-Robison et al., 2011; Mcclung et al., 2005).
The biophysical mechanisms underlying the increased ex-
citation that appears in a DA cell with an undersized struc-
ture is currently poorly understood. In order to explain the
latter, most researchers will make use of the well-established
experimental rule that the augmentation of the membrane
conductance to K+ inhibits cell activity (Arencibia-Albite
et al., 2007), while its reduction enhances excitability (Hopf
et al., 2007; Ji et al., 2012). In view of this empirical prin-
ciple, Mazei-Robison et al. (Mazei-Robison et al., 2011)
showed that K+ channel expression is down-regulated in
morphine-exposed mice and concluded that such an event is
at least one of the main culprits for the augmented cell firing.
As a consequence, if this reasoning is correct then the ap-
propriate homeostatic response that recovers normal intrinsic
excitability is the return of each altered conductance to its
natural pretreatment state, that is, the size of each distinct
ion channel population in the neuronal membrane should be
equal to that of the untreated condition. The present report
uses numerical simulations to show that, surprisingly, this
is not the case. In striking contrast to experimental reason-
ing, the computational modeling illustrates that the reduc-
tion in K+ conductance, in the presence of diminished cell
size, is a compensatory mechanism that dampens the aug-
mented firing and not an event that drives further excitation
as qualitative inspection of experimental data may suggest.
Simulations also illustrate that a smaller DA neuron will ex-
press a reduced synaptic summation index; yet, it will burst
at a much faster rate and with a higher spike count per burst
than a normal size cell when exposed to the same train of
excitatory post-synaptic currents.
2
|
MATERIALS AND METHODS
The model used in this study is adapted from existing models
of DA neurons (Komendantov et al., 2004; Kuznetsova et al.,
2010; Yu & Canavier, 2015; Yu et al., 2014). Komendantov
et al. (Komendantov et al., 2004) represent the typical DA
neuron with thirteen compartments, including one soma, four
proximal dendrites, and eight distal dendrites. Symmetry
considerations allow the thirteen-compartment model to be
equivalent to a three-compartment model: one soma, one
proximal dendrite, and one distal dendrite (Kuznetsova et al.,
2010; Yu et al., 2014). This not only simplifies the model's
computational implementation but also captures the funda-
mental electrophysiological properties of multi-compart-
mental models that expressed a realistic DA cell morphology
(Kuznetsova et al., 2010). The DA neuron model used in this
study consists, therefore, of these three compartments.
The model compartments are treated as cylinders with de-
fined length (L) and diameter (d) whose equivalent circuits
are as shown in Figure 1a. The current balance equations for
compartments are (S = soma, P = proximal dendrite, D =
distal dendrite):
C
S
dV
S
dt=−INa,S −IK,S −ISK,S −IA,S −IMU,S −I
GIRK,S
−I
CaL,S
−I
h,S
−I
L,S
+g
SP (
V
P
−V
S)
C
P
dV
P
dt=−INa,P −IK,P −ISK,P −IA,P
−IMU,P −IGIRK,P −ICaL,P −Ih,P −I
L,P
+g
SP (
V
S
−V
P)
+g
PD (
V
D
−V
P)
C
D
dV
D
dt=−INa,D −IK,D −ISK,D −IA,D −IMU,D −I
GIRK,D
−I
CaL,D
−I
h,D
−I
L,D
+g
PD (
V
P
−V
D)
|
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ARENCIBIA-ALBITE ANd JIMÉNEZ-RIVERA
where INa,X is the fast Na+ current, IK,X is the delayed rectifier
K+ current, ISK,X is the small K+ conductance current, IA,X is the
transient outward K+ current, IMU,X the muscarinic K+ current,
IGIRK,X is the G-protein-gated inward rectifying K+ current,
ICal,X is the non-inactivating L-type Ca++ current, Ih,X is the hy-
perpolarization-activated cation current, IL,X is the leak current,
gSP(VP− VS) and gSP(VS−VP) are the coupling currents be-
tween the soma and the proximal dendrite, while gPD(VD−VP)
and gPD(VP−VD) couple the proximal and distal dendrites. The
length, diameter, surface area (Sa), and capacitance (C) for each
compartment are:
L
S=25 𝜇m,dS=15 𝜇m, SaS≈1, 178 𝜇m
2,
CS
=20 pF
(
1.7 𝜇F∕cm2
)
L
P=150 𝜇m,dP=3𝜇m, SaP≈1, 414 𝜇m
2,
CP
=30pF
(
2.122 𝜇F∕cm2
)
FIGURE 1 DA neuron model electrophysiology. (a) All compartments in the model are equivalent to the depicted circuit. Na+ and Ca++
currents are inward whereas K+ currents are outward. The h and leak currents reverse direction during the action potential time course. For further
details see Materials and Methods. (b) The model's soma compartment exhibits a slow pacemaker firing at 2.16Hz under control conditions. (c)
Measurement of the model's membrane time constants when the soma compartment is held at −70mV. (d1) Input resistance test performed at the
soma compartment. (d2) The graph expands the content of the dashed box in c1. (e) The model's burst response as elicited by NMDA channels
placed at the soma compartment. (f) The soma spontaneous activity in the model is driven by the L-type Ca++ conductance (gCaL); gCaL was set to
zero in all compartments. (g) The inhibition of the h-conductance (gH) has no impact on the model's pacemaker activity; gH was set to zero in all
compartments. (h) Depolarization sag response measured at the soma compartment. (i1) Simulated voltage-clamp whole-cell recording. The pipette
was placed at the soma compartment. The access resistance is 30MΩ. (i2) Zoom in on the recording in i1.
4 of 18
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Ionic currents in the model obey the following set of
equations. Parameters values in the equations are listed as
they appear in the numerical algorithm. The units in the pro-
grammed iterations were mV for membrane potential, nS for
conductance, pA for current, and ms for time.
Fast Na+ current (INa):
Delayed rectifier K+ current (IK):
Small K+ conductance current (ISK):
where fCa = 0.001; F = 96,520 C/mol; vol = 0.0117 fL;
βCa=0.05; [Ca++]min=0.00001mM.
Transient outward K+ current (IA):
Muscarinic K+ current (IMU):
G-protein gated inward rectifying K+ current (IGIRK):
L
D=350 𝜇m,dD=1.5 𝜇m, SaD≈1, 649 𝜇m
2,
CP
=30 pF
(
1.82 𝜇F∕cm2
)
I
Na,X =gNa,X m
3
ph
SS(VX−ENa
),
g
Na,S =gNa,P =gNa,D =450 nS
ENa =40 mV
dm
dt
=𝛼m
(
VX
)
(1−m)−𝛽m
(
VX
)m
dp
dt
=𝛼p
(
VX
)
(1−p)−𝛽p
(
VX
)p
d
hSS
dt=hSS,∞
(
VX
)
−hSS
𝜏
hSS (
V
X)
𝛼p(
V
X)
=0.07 exp
[
−
(
V
X
+40
)
∕20
]
𝛼
m
(
VX
)
=
0.1V
X
+2.5
1−exp
[
−
(
0.1V
X
+2.5
)]
𝛽
p
(
VX
)
=
1
1+exp
[
−
(
0.1V
X
+1.4
)]
𝛽m(
V
X)
=4exp
[
−
(
V
X
+50
)
∕18
]
h
SS,∞
(
VX
)
=
1
1+exp
[
V
X
+45
]
𝜏
hSS
(
VX
)
=20 +
580
1+exp
[
V
X]
I
K,X =gK,X n
4(
VX−EK
)
,gK,S =
225 nS,
gK,P
=g
K,D
=175 nS E
K
=−78 mV
dn
dt
=𝛼n
(
VX
)
(1−n)−𝛽n
(
VX
)n
𝛼
n
(
VX
)
=0.01
(
VX+34
)
1−exp
[
−0.2
(
V
X
+34
)]
𝛽n(
V
X)
=0.125exp
[
−
(
V
X
+40
)
∕80
]
I
SK,X =gSK,X
VX−EK
1+KSK∕Ca++ in,X4,
g
SK,S =gSK,P =0.25 nS, gSK,D =
0.3 nS
KSK =0.00019 mM
d[
Ca
++]
in,X
dt
=−
fCaICaL,X
2F⋅vol
−𝛽Ca
([
Ca++
]
in,X −
[
Ca++
]
min
)
IA,X
=g
A,X
rq
3(
V
X
−E
K)
,g
A,S
=3 nS, g
A,P
=g
A,D
=
4 nS
d
r
dt
=r∞
(
VX
)
−r
20
d
q
dt
=q∞
(
VX
)
−
q
15
r
∞
(
VX
)
=
1
1+exp
[(
V
X
+63∕4
)]
q
∞
(
VX
)
=
1
1+exp
[
−
(
V
S
+43
)
∕24
]
I
MU,X =gMU, X 𝜇
(
VX−EK
)
,
gMU,S
=1.5 nS, g
MU,P
=1.8 nS, g
MU,D
=
2.1 nS
d𝜇
dt
=𝛼μ
(
VX
)
(1−𝜇)−𝛽μ
(
VX
)𝜇
𝛼
μ
(
VX
)
=
0.02
1+exp
[
−
(
V
X
+20
)
∕5
]
𝛽μ(
V
X)
=0.01exp
[
−
(
V
X
+43
)
∕18
]
I
GIRK,X =gGIRK,X
(
VX−EK
)
1+exp
[(
VX+45
)
∕20
]
,
gGIRK,S
=0.012 nS, g
GIRK,P
=0.0144 nS, g
GIRK,D
=
0.0168 nS
|
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L-type Ca++ current (ICaL):
Hyperpolarization-activated cation current (Ih):
Ether-a-go-go related gen K+ (ERG) current (IERG):
Leak currents (IL):
The coupling conductances between compartments are:
2.1
|
Simulations in cell size alteration
In order to simulate reductions in cell size we assumed, as a
first approximation, that a neuron with a decreased size pre-
serves its relative dimensions. Thus, as cell capacitance de-
creases the ratio of length to diameter in each compartment
is kept constant. This signifies, as an example, that if cell
capacitance is decreased by 36% the coupling conductances
between compartments are adjusted using the following new
diameter and length values
Since
where
Cm
is the specific membrane capacitance.
2.2
|
Simulations of afferent driven activity
AMPA and NMDA excitatory synaptic currents were placed
at the distal dendrite compartment and modeled by the fol-
lowing equations:
I
CaL,X =gCaL,Xl
(
VX−ECa
)
,
g
CaL,S =0.14875 nS, gCaL,P =0.2125 nS, gCaL,D =
0.2975 nS,
ECa =70 mV
d
l
d
t=l∞
(
VX
)
−
l
𝜏
l(
V
X)
l
∞
(
VX
)
=
1
1+exp
[
−
(
V
X
+42
)
∕12
]
𝜏
l
(
VX
)
=5exp
[
−
(
VX+70
)
2∕625
]
+
0.25
Ih,X =gh,X hX
(
VX−Eh
)
;
gh,S =2.5 nS, gh,P =3 nS gh,D =
3.5 nS,
Eh=−53 mV
d
hX
dt=h∞,X
(
VX
)
−hX
𝜏
h,X (
V
X)
h
∞,X
(
VX
)
=
1
1+exp
[(
V
X
+90
)
∕8
]
𝜏
h,X
(
VX
)
=425exp
(
0.075
(
VX+112
))
1+exp
(
0.083
(
V
X
+112
))
IERG,X =gERG,Xo
(
VX−EK
)
;
g
ERG,S
=0.3 nS, g
ERG,P
=0.36 nS g
ERG,D
=
0.42 nS
do
dt
=𝛼o(1−o−i)+𝛽ii−o
(
𝛼i+𝛽o
)
di
dt
=𝛼io−𝛽i
i
𝛼o(
V
X)
=0.0036exp
(
0.0759V
X)
𝛽o(
V
X)
=1.2523 ×10
−5
exp
(
−0.0671V
X)
𝛼i(
V
X)
=92.11exp
(
0.1189V
X)
𝛽i(
V
X)
=12.6exp
(
0.0733V
X)
I
L,X =gL,X
(
Vx−EX
)
,
g
L,S =0.35 nS, gL,P =gL,D =
0.65 nS,
ES=EP=ED=−58 mV
g
SP =𝜋×10
−4
2Ri[LS
d2
S
+LP
d2
P]=234 nS,
g
PD =𝜋×10−4
2Ri[LP
d2
P
+LD
d2
D]=
22.8 nS,
Ri=
40
Ω−
cm
L
new =
�
1−36
100
Lold =
√
0.64Lold =0.8L
old
d
new =
�
1−36
100
dold =
√
0.64dold =0.8d
old
C
compartment,new =
(
1−36
100
)
Cm⋅Sacompartment,old
=(1−36
100 )
Cm𝜋doldLold
=
Cm𝜋
{√
1−36
100 dold
}{√
1−36
100 Lold
}
I
AM PA =gAMPA rA MPA
(
Vm−EAM PA
);
gAM PA
=4nS,
EAM PA
=0mV
I
NMDA =gN MDArN MDA
(
Vm−ENMDA
)
1+exp
(
−0.062Vm⋅
[
Mg
]
∕3.57
)
;
gNMDA
=15 nS, E
NMDA
=0 mV,
[
Mg
]
=
1.5 mM
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Glutamate release was modeled by the following step
function
where [GLU] is the glutamate concentration at the synaptic
cleft.
2.3
|
Numerical integration
To avoid numerical stability issues the derivates in the equa-
tions where approximated by backward finite differences
obtaining the following matrix equation
where
G
S=
X
gX,S
VS,t
,GP=
X
gX,P
VP,t
,GD=
X
gX,D
VD,t
.
As the above coefficient matrix is non-singular the latter
equation was solved by matrix inversion. The numerical iter-
ations were programmed and run in Mathematica 10.4 soft-
ware using a time step of 0.1ms.
3
|
RESULTS
3.1
|
The model captures the characteristics
features of DA cell electrophysiology
Figure 1 depicts the model behavior. The spike count for
a simulation time length of 6 seconds is 13 (Figure 1b).
Injecting a constant current of −35.1299 pA to the soma
compartment clamps the membrane potential to −70 mV,
while the membrane time constant relaxes to around 28.4ms
(Figure 1c). At this potential, the model's input resistance is
near 278.7MΩ (Figure 1d1,2). Placing a 20nS NMDA con-
ductance at the model's soma elicits a burst response similar
to that recorded during dynamic clamp experiments in the
midbrain slice preparation (Figure 1e) (Lobb et al., 2010,
2011). As in other DA neuron models (Komendantov et al.,
2004; Kuznetsova et al., 2010), what drives the pacemaker
activity is the L-type Ca++ conductance with no contribution
of the h-current (Figure 1f,g). Although in mice Ih appears to
contribute to the spontaneous activity (Okamoto et al., 2006),
in the rat brain slice this remains controversial. Here we used
an Ih voltage dependence and reversal potential similar to that
of Hopf et al. (Hopf et al., 2007), where bath application of
the h-channel blocker ZD7288 did not alter intrinsic DA cell
activity. Notice that in the model although Ih does not in-
fluence spontaneous spiking the h-conductance response to
membrane hyperpolarization is robust (Figure 1h, i1, and i2).
3.2
|
Effects of cell size alterations on
intrinsic activity
A reduction in neuronal size implies a diminished cell sur-
face area, which may result in a lower value of membrane
conductance since now there is less membrane to insert ion
channels. Cell capacitance (Cm) will also be decreased as this
parameter is directly proportional to the membrane surface
area. All simulations in this work regarding the effects of size
alterations are, therefore, executed by modifications in Cm
similar to that observed in rodent models of drug addiction
(see Materials and Methods) (Arencibia-Albite et al., 2012;
Mazei-Robison et al., 2011).
In morphine-exposed mice, a reduction in DA cell size,
that shrinks the mean cell surface area by nearly 30%, is ac-
companied by a substantial reduction in the expression of
various K+ conductances plus a significant increment in au-
tonomous spiking (Mazei-Robison et al., 2011). This suggests
that in order to normalize the excitability of the affected cell it
is necessary to return the magnitude of all distinct populations
of ion channels to its pre-morphine level. Or equivalent, a DA
cell that diminishes its size but conserves the pre-treatment
dr
AM PA
dt=𝛼AM PA [GLU]
(
1−rAM PA
)
−𝛽AM PA rAMPA
;
𝛼AM PA =1.1 (mM
⋅
ms)
−1
,
𝛽
AM PA =0.19 ms
−1
dr
NMDA
dt=𝛼NMDA [GLU]
(
1−rNMDA
)
−𝛽NMDA rNMDA
;
𝛼NMDA =0.072 (mM
⋅
ms)
−1
,
𝛽
NMDA =0.0066 ms
−1
[
GLU]=
{
1 mM ton ≤t≤ton +
1ms
0 mM otherwise
d
VX
dt
≈V
(t)
X−V
(t−Δt)
X
Δt
1+
GS+gSP
Δt∕CS
−gSPΔt∕CS0
−gSPΔt∕CP1+GP+gSP +gPD Δt∕CP−gPDΔt∕CP
0−gPDΔt∕CD1+GD+gPD Δt∕CD
V(t)
S
V(t)
P
V(t)
D
=
V(t−Δt)
S+ESΔt∕CS+I(t)
CLAMP
V(t−Δt)
P+EPΔt∕CP
V(t−Δt)
D+EDΔt∕CD
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number of all membrane ion channels should maintain an
unaltered spontaneous activity. We next test the validity of
this claim by reducing the capacitance of the model neuron in
the presence of an unchanged conductance, that is, the peak
magnitude of each conductance is kept constant in all com-
partments throughout all simulations.
Figure 1b shows the control condition where the model
generates 13 spikes in 6seconds. In Figure 2a, a 30% re-
duction in cell capacitance increases the spike count to
17. Likewise, a reduction of 50% (Figure 2b) and (Figure
2c) 80% results in spike counts of 20 and 28, respec-
tively. Figure 2d summarizes simulations and shows that
a reduction in whole-cell capacitance, in the presence of
a non-changing number of ion channels, increases the in-
trinsic activity (black dots). The reduction of the somatic
capacitance alone, in contrast, evoked non-significant incre-
ments in spontaneous firing (white dots). These simulations
illustrate, consequently, that conserving a constant number
of ion channels does not prevent the increase in spiking fre-
quency that follows from a reduction in cell size.
Figure 2e illustrates that a reduced cell capacitance in the
presence of a constant peak conductance level implies a sub-
stantial relative increment in ion channel density (i.e., number
of ion channels per unit of membrane surface area). This sig-
nifies that, as a homeostatic response, the conservation of a
constant number of ion channels represents a futile adaptation
to cell size diminution since the metabolic cost that under-
lies the elevated density does not halt the increase in spiking
frequency. The next section shows how the channel density
scaling, which results from a reduced cell size, affects intrin-
sic activity.
3.3
|
Effects of ion channel density scaling
on intrinsic activity
In the next simulations as the cell capacitance is decreased
by 30% the peak magnitude of all membrane conductances
(denoted by Gm) will be unchanged, increased, or de-
creased relative to its pre-reduction magnitude. These com-
putational experiments illustrate how the intrinsic activity
responds to changes in the net current density time course
that are induced by alterations in ion channel densities. In
these numerical recreations when Gm is altered by a fixed
proportion, the peak magnitude of each conductance in all
compartments has been scaled by the same proportion. We
FIGURE 2 Effects of DA cell size
attenuation in the presence of a fixed
number of membrane ion channels. (a) A
Cm reduction of 30% elevates cell firing by
~31% (from 2.16 to 2.83Hz) and increases
the action potential amplitude (APA) by
5.98mV (from 98.46 to 104.44mV).
(b) A Cm reduction of 50% elevates cell
firing by ~54% (from 2.16 to 3.33Hz)
and increases the APA by 9.86mV (from
98.46 to 108.32mV). (c) A Cm reduction
of 80% elevates cell firing by ~115% (from
2.16 to 4.66Hz) and increases the APA
by 14.98mV (from 98.46 to 113.44mV).
(d) Scatter plot that summarizes the effects
of cell size alterations in the presence of a
fixed number of membrane ion channels.
(e) Scatter plot showing that a decline in
Cm, under a fixed number of membrane ion
channels, results in a substantial elevation
of the densities of each channel type. For
example, a 30% drop in capacitance elevates
densities by about 43%.
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proceeded in this manner for two reasons. First, it is un-
likely that a change in intrinsic activity, secondary to cell
size reduction, is the result of alterations in exactly a single
conductance type. Second, in order for the DA cell to con-
serve its pacemaker activity, after size diminution, the rela-
tive change in the magnitude of outward currents cannot be
much greater or much smaller than that of inward currents
since both situations may lead to the abolition of spontane-
ous spiking. Therefore, in order for the intrinsic activity to
survive an event of substantial size reduction, the variabil-
ity among the different relative changes ascribed to each
conductance has to be small. A reasonable approximation
to such a case is to assume uniform scaling among the dis-
tinct membrane conductances. This approach, as shown
next, clearly illustrates how intrinsic activity adjusts to
modifications in ion channel density.
Panels a and b in Figure 3 illustrate that when a drop in
cell capacitance is followed by a rise in Gm the spike count
FIGURE 3 Effects of channel density scaling secondary to DA cell size reduction. (a) A reduction of 30% in Cm with a 15% increase in Gm
elevates the channel density of each conductance by ~64%, in contrast, to control (Figure 1b). In this case, the spike count has augmented by five
spikes (from 13 to 18; 2.16Hz → 3Hz). Notice here that gK has augmented by 15% yet cell firing is enhanced. (b) A reduction of 30% in Cm with
a 30% increase in Gm elevates the channel density of each conductance by ~86%, in contrast, to control (Figure 1b). In this case, the spike count has
augmented by seven spikes (from 13 to 20; 2.16Hz → 3.33Hz). Notice here that gK has raised by 30% yet cell firing is enhanced. (c) A reduction
of 30% in Cm with a 15% decrease in Gm elevates the channel density of each conductance by ~21%, in contrast, to control (Figure 1b). In this case,
the spike count has augmented by two spikes (from 13 to 15; 2.16Hz → 2.5Hz). When contrasted to Figure 2a, the 15% declined in gK correlates
with cell excitation dampening and not with enhance cell firing. (d) A reduction of 30% in Cm with a 30% decrease in Gm conserves the channel
density of each conductance. In this case, the spike count and the membrane voltage trace remain identical to control (Figure 1b). When contrasted
to parts a, b, and c, the 30% declined in gK correlates with the preservation of cell excitation and not with enhance cell firing. (e) A reduction of
30% in Cm with a 50% decrease in Gm decreases the channel density of each conductance by ~29%, in contrast, to control (Figure 1b). In this case,
the spike count has declined by three spikes (from 13 to 10; 2.16Hz → 1.66Hz). When contrasted to parts a, b, c, and d, the 50% declined in gK
correlates with cell excitation dampening and not with enhance cell firing. (f) Scatter plot that summarizes the above simulations (black dots). The
horizontal axis has been label as gK, and not Gm, to emphasize that a decrease in gK, after size reduction, correlates with cell excitation dampening
and not with enhance cell firing. A decrease in gK increases cell excitation only when all other cell properties are held constant (white dots).
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also increases. Panels c–e show, in contrast, that a decline in
Gm rescues the intrinsic activity only if the relative change in
Gm matches that in Cm. Consequently, is possible to conserve
the spiking frequency, after cell size attenuation, by just pre-
serving the channel density for each conductance present in
the membrane.
Figure 3 also challenges a widespread principle in neu-
roscience: “a drop in K+ conductance augments cell exci-
tation.” Panel f summarizes all simulations in Figure 3 and
shows, for example, that under a reduced cell size a decline
in the net K+ conductance (gK) tends to depress intrinsic ac-
tivity and not to enhance it as suggested by this empirical
norm. As cell size diminishes there is less membrane area to
insert ion channels and hence the net conductance for each
permeable ion is likely to be decreased. As a result, when
other membrane elements are changing is not possible to pre-
dict alterations in cell excitation by simply inspecting if gK
has rise, not change or decay. In effect, in Figure 3d gK has
decreased by 30% which by itself it is expected to increase
cell excitation, yet as seen in the simulation the intrinsic ac-
tivity remains unchanged relative to the control case. Thus, a
reduction in gK correlates with increase excitation only when
all other biophysical properties are held constant (white dots
in Figure 3f).
3.4
|
The ether-a-go-go related gen K+
current may prevent the increased DA cell
firing secondary to size reduction
A decline in cell size diminishes cell surface area leading to
alterations in channel densities. If the tendency, subsequent
to size attenuation, is to raise the membrane concentration
of all ion channels then the autonomous firing frequency
is likely to be elevated (see Figure 3a,b). Is possible, as
previously discussed, to rescue the normal intrinsic activity
by conserving the original ratios of the number of chan-
nels per unit of area (see Figure 3d). This section reveals,
nonetheless, a remarkable and unexpected biophysical
mechanism that effectively buffers the increased DA cell
excitation secondary to size reduction. Such a mechanism
does not require the preservation of the channel density for
each conductance.
DA neurons from the rat express the ether-a-go-go related
gen K+ (ERG) current termed IERG (Canavier et al., 2007; Ji
et al., 2012). In these cells, the selective blockade of ERG
channels increased the frequency of spontaneous activity
as well as the firing response to current injection and also
accelerates the entry into depolarization block as evoked by
dynamic clamp bursting (Ji et al., 2012). These properties, as
explained next, result from the IERG activation kinetics and
voltage dependence.
ERG channels activate slowly with depolarization but,
as the amplitude of stimulus increases, inactivation oc-
curs almost immediately after channel opening (Ficker
et al., 2001). Therefore, during the upward phase of the
somatic spike, a substantial proportion of ERG channels
have transitioned into the inactive state. During the down-
ward phase, as inactive ERG channels return to the close
state, they must pass first through the open conformation.
The acquisition of the open state is nearly instantaneous;
however, open ERG channels recover the close state at a
much slower rate resulting in a significant outward cur-
rent between consecutive action potentials. Consequently,
ERG channels limit cell firing by providing a robust in-
terspike outward current that dampens the intensity of
depolarizing currents.
Figure 4a shows the incorporation of the ERG conduc-
tance into the computational model. In accordance with
experimental data (Ji et al., 2012), the inhibition IERG in
the model increases intrinsic activity (Figure 4b). The
activation curve of the ERG channel demonstrates that
at voltages close to the peak of the action potential the
fraction of open channels is non-significant (Figure 4c1).
The IERG IV curve shows that ERG channel inactivation
increases as the amplitude of the command potential in-
creases (Figure 4c2). Yet, as the repolarization amplitude
augments, tail currents become larger and last longer
(Figure 4c2). Consequently, if during cell size reduc-
tion the ERG channel density increases the rise in firing
frequency could be significantly dampened. This claim
is reasonable for two reasons. First, a reduced cell size
increases the amplitude of action potentials (see Figure
2) which, in turn, augments the fraction of inactive ERG
channels. Second, on repolarization, a large inactive frac-
tion leads to a large open fraction that when combined
with an increased channel density enhances IERG density.
This may slow down the elevated depolarizing drive that
emerges with diminished cell morphology.
We next address the latter hypothesis by repeating
simulations in Figure 3 but now in the presence of IERG.
Figure 4a is the control condition, and as expected, a re-
duction of 30% in cell capacitance, while Gm remains
constant (Figure 4d) or increased up to 30% (Figure 4e),
augments the spike count by just one spike. Notice also
that the action potential waveform and number remain
identical to control only when the relative drop in Gm is
equal to that in cell capacitance (Figure 4f). Black dots
in Figure 4g summarize simulations and suggest that
in the presence of IERG cell size attenuation is unable
to evoke a substantial elevation in the intrinsic firing.
Figure 4g also emphasizes, once again, that when other
membrane elements are changing is not possible to pre-
dict alterations in cell excitation by just examining the
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expression level of gK. A reduction in gK increases cell
excitation only when all other membrane properties are
clamped (white dots in Figure 4g). Figure 5 substantiates
further the addressed hypothesis as it illustrates in-
stances where IERG amplitude increases with decreasing
cell size.
FIGURE 4 The ether-a-go-go related gen K+ (ERG) current may prevent the increased DA cell firing secondary to size reduction. (a) The
initial placement of the ERG conductance in the DA neuron model breaks the balance between inward and outward currents abolishing intrinsic
activity (gray trace). After placing the ERG conductance in all compartments, the control firing frequency is recovered by increasing the non-
inactivating L-type Ca++ conductance (gCaL). (b) The blockade of IERG doubles the firing frequency (from 2.16 to 4.33Hz); the ERG conductance
(gERG) was set to zero in all compartments. (c1) ERG channel activation curve. (c2) I–V curve of the somatic IERG. Holding potential was −70mV.
Step commands were from −50 to 50mV in 20mV increments. The red trace is the current response to the 50mV step. (d) In the absence of
IERG, a 30% reduction in Cm, with no change in Gm, increases the spike count by four spikes (from 13 to 17; 2.16 → 2.83Hz, see Figure 2a). In the
presence of IERG, however, the spike count increases by one spike (from 13 to 14; 2.16 → 2.33Hz). (e) In the absence of IERG, a 30% reduction in
Cm, with a 30% increase in Gm, increases the spike count by seven spikes (from 13 to 20; 2.16 → 3.33Hz, see Figure 3b). In the presence of IERG,
however, the spike count increases by one spike (from 13 to 14; 2.16 → 2.33Hz). (f) A reduction of 30% in Cm together with a 30% decrease in
Gm conserves the spike count. In this case, the membrane voltage trace remains identical to control (Figure 4a). (g) Scatter plot that summarizes
the simulations in the presence of IERG (black dots). The horizontal axis has been label as gK, and not Gm, to emphasize that when other biophysical
attributes are changing is not possible predict the adjustments in cell firing by just measuring the expression level of gK; gK is inversely related to
cell excitation only when all other membrane properties are held constant (white dots).
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3.5
|
Effects of cell size attenuation on
afferent-driven activity
The DA neuron model is next current-clamped at −55mV
and excited at the distal dendrite. Intermittent synaptic exci-
tation onto the latter compartment is simulated by a train of
ten 1mM glutamate pulses (duration: 1 ms) at a frequency
of 33⅓Hz. Such stimulus evokes the transient activation of
AMPA and NMDA ligand-gated channels resulting in the
somatic summation of 10 excitatory post-synaptic potentials
(EPSPs). The maximal values of these dendritic conduct-
ances are fixed in all simulations.
Figure 6a1 illustrates the control case where the
train stimulus triggers a burst of 6 spikes. The EPSPs
integration that underlies this response can be ob-
served by setting the Na+ conductance (gNa) to zero
in all compartments (Figure 6a1,a2). Here the tem-
poral summation index (TSI) is given by the relative
change in the last EPSP with respect to the first, that is,
TSI
=
(
EPSP
last
∕EPSP
first
−1
)
×
100
. Our previous work
(Arencibia-Albite et al., 2017) has shown that, under cell
size attenuation, the TSI may be a poor indicator of the
quality of synaptic integration. We proposed that to better
assess the level of temporal summation, the TSI has to
FIGURE 5 IERG amplitude increases in response to cell size reduction. (a1) In the absence of IERG, a reduction of 50% in Cm, with no change
in Gm, increases the spike count by seven spikes (from 13 to 20; 2.16 → 3.33Hz, see Figure 2b). In the presence of IERG, however, the spike count
increases by one spike (from 13 to 14; 2.16 → 2.33Hz). The APA was increased by 9.64mV (from 98.46 to 108.1mV). (a2) The increment in
APA measured in a1 increases IERG activation (black trace vs. red trace). The somatic IERG mean value was augmented by ~22% (from 0.91 to
1.11pA). Such enhanced amplitude was able to effectively dampen the expected raise in intrinsic firing after a 50% Cm reduction. IERG mean value
is defined as
‼
I
ERG =1
6
∫6
0IERG d
t
. The Vm gray trace is identical to the Vm trace in a1. (a3) The graph expands the content in a2 in the shown time
interval. (b1) In the absence of IERG, a reduction of 80% in Cm, with no change in Gm, increases the spike count by 15 spikes (from 13 to 28; 2.16 →
4.66Hz, see Figure 2c). In the presence of IERG, however, the spike count remains constant. The APA was increased by 14.93mV (from 98.46 to
113.39mV). (b2) The APA increment measured in C1 increases the IERG mean value by ~37% (from 0.91 to 1.25pA). This elevated amplitude was
sufficient to prevent the expected raise in intrinsic firing after an 80% Cm reduction. The Vm gray trace is identical to the Vm trace in b1. (b3) The
graph expands the content in b2 in the shown time interval.
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be accompanied by the computation of the average mem-
brane depolarization (ΔVavg); defined as
where ΔVm is the change in membrane potential with re-
spect to the holding voltage and T represents the time span
between the peak values of the first and last EPSP. As TSI
increases ΔVavg is expected to increases since these mea-
surements are usually positively correlated (Angelo et al.,
2007; Brager & Johnston, 2007; Carr et al., 2007; Lewis
et al., 2011). Therefore, if cell size reduction improves
afferent-driven activity both measurements should augment
after size attenuation.
Figure 6b1,b2 illustrate the model response when Cm and
Gm are decreased by 30% and 15%, respectively. Notice that
the enhanced burst signal results from the augmented ΔVavg
yet TSI has diminished by 38%. Likewise, decreasing Gm by
30% (Figure 6c1) or 50% (Figure 6d1), in the presence of a
30% drop in Cm, further increases the spike count (Figure
6c1,d1). The TSIs in these cases are also reduced, while
the respective ΔVavg values are increased (Figure 6c2,d2).
Altogether it suggests that diminished cell size is associ-
ated with a reduced TSI that paradoxically correlates with
an elevated ΔVavg and hence an improved afferent-driven
activity; similar to what we have shown in an animal model
Δ
Vavg =1
T
T
∫
0
ΔVmd
t
FIGURE 6 DA cell size reduction
results in a paradoxical enhancement of the
afferent-evoked burst signal. AMPA (4nS)
and NMDA (15nS) channels were placed
in the distal dendrite. This compartment
was then excited by a train of 10 glutamate
squared pulses. Simulations in this figure
include the ERG conductance in all
compartments as no substantial differences
were observed in its absence (data not
shown). (a1) Control burst signal at the
soma compartment. The EPSP summation
that elicits this response is exposed by
setting gNa=0 in all compartments (dashed
curve). Cm and Gm were held fixed. (b1)
Burst signal after Cm and Gm are decreased
by 30% and 15%, respectively. (c1) Burst
signal after Cm and Gm are both decreased
by 30%. (d1) Burst signal after Cm and Gm
are decreased by 30% and 50%, respectively.
a2, b2, c2, d2. Zoom in on the temporal
summation recording in part 1. In c2,
b2, and d2, the TSI has experienced a
substantial depression relative to that of
the control recording (a2), yet the average
membrane depolarization has increased.
Such behavior represents a paradoxical
response according to the current view of
the TSI. Notice that, in all figures relative
to a1, the elevated average membrane
depolarization is what underlies the reduced
inter-spike interval and elevated spike count.
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of chronic cocaine administration (Arencibia-Albite et al.,
2017).
4
|
DISCUSSION
4.1
|
The positive correlation, between firing
frequency and input resistance, only holds
when the cell size is fixed
A reduced DA cell size, induced by morphine exposure or
by Clock gene mutation, correlates with an increased au-
tonomous firing (Coque et al., 2011; Mazei-Robison et al.,
2011; Mcclung et al., 2005). This finding is believed to
be primarily the result of a reduction in K+ channel ex-
pression (Mazei-Robison et al., 2011); which, as argued by
other theoretical studies, increases the intrinsic spiking rate
as a consequence of the rise in the cell's input resistance
(Enrico et al., 2016; Sengupta et al., 2013). If this analysis
is correct then the appropriate homeostatic response that
restores spontaneous activity is the conservation of the pre-
treatment population size for each ion channel type; which
signifies an elevated channel density for each distinct mem-
brane conductance. Although such adaptation preserves the
input resistance it did not prevent the raise in cell firing as
shown here. Our computational experiments illustrate, in
contrast, that the response that conserves the pretreatment
activity is the preservation of the channel density of each
conductance, which elevates the input resistance as cell
size diminishes. Consequently, is likely that the down-reg-
ulation of gK, under cell size attenuation, dampens the raise
in intrinsic activity by decreasing the K+ channels density.
This furthers implies that the proposed positive correlation
between firing frequency and input resistance, only holds
when the cell size or capacitance is kept fixed; otherwise,
is not possible to infer the change in cell firing by only
measuring the change in input resistance.
4.2
|
The conservation of ion channels
densities, after cell size reduction, preserves the
rate and pattern of spontaneous firing
The Appendix section explains in detail how the conservation
of the channel density of each conductance preserves the rate
and pattern of spontaneous firing. The given analytical argu-
ment also elucidates why a constant number of ion channels
does not halt the rise in intrinsic firing associated with a de-
crease in cell morphology. The biophysical interpretation of the
argument is remarkably simple: if the number of channels is
kept constant then, under cell size reduction, the inward and
outward currents will be elevated per unit of membrane surface
area and thus the time required to charge and discharge the now
smaller cell membrane will be substantially diminished. Such
a process is very similar to the problem of how many times,
during a fixed time period, a water tank can be filled and emp-
tied: If the inflow and outflow rates are kept constant then the
number of fill-empty cycles will increase when the tank size is
reduced. This is in essence what explains the augmented firing
frequency as cell size decreases, while the number of ion chan-
nels remains clamped.
4.3
|
Somatic size reduction has minimal
impact on spontaneous cell activity
Enrico et al. (Enrico et al., 2016) had implemented a realistic
DA neuron model of 28 compartments (soma, 11 proximal
dendrites and 16 distal dendrites) to analyze the biophysical
effects of opiates withdrawal. In their work, size reduction
was restricted to the somatic compartment since as explained
by these authors morphine effects on the structure of the den-
drites are not clear. Modifications in cell firing after somatic
shrinkage were, however, non-substantial. In agreement with
this finding, cell spiking in our model also experienced minor
changes when size contraction was limited to only somatic
dimensions. It seems, therefore, that in order for a given mor-
phological alteration to impact cell activity it needs to affect
a significant proportion of the whole-cell surface area; if not
the effects appear to be inconsequential. As a result, the pre-
sent theoretical study suggests that in order for morphine ex-
posure or Clock gene mutation to evoked a significant raise
in the spontaneous activity it has to also reduced a significant
proportion of the dendrites surface area. Further experimen-
tation is required to test such a prediction.
4.4
|
The presence of the ERG current may
limit the rise in spontaneous activity after cell
size reduction
In rats repeated cocaine administration results in a sub-
stantial decline in the whole-cell capacitance of DA cells
(Arencibia-Albite et al., 2012, 2017). In contrast to mice,
the spontaneous activity remains apparently unaltered after
capacitance reduction (Arencibia-Albite et al., 2012). A
possible explanation for this discrepancy could be the pres-
ence of the ERG current. In rats the ERG conductance is
highly expressed and is known to decrease the level of in-
trinsic cell firing by providing a strong interspike outward
current (Ji et al., 2012). The outstanding aspect of this K+
current is that the open fraction of ERG channels increases
as the spike's repolarization rate rises (see Figure 4c2).
Thus, is possible, that the smaller capacitance that elicits a
faster spike is also leading to a level of channel recruitment
that exceeds the pre-reduction open number; even if the
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FIGURE 7 A hypothesis that experimental evidence seems to validate may be shown to be erroneous by numerical simulation. Chronic
morphine exposure results in smaller midbrain dopamine neurons that express an elevated spontaneous firing which as argues by qualitative
reasoning is simply a consequence of the rise in the cell's input resistance (RN) after size reduction. Motivated by this apparently reasonable
hypothesis, experimental studies have shown that concomitant to this increased excitation gK expression is substantially diminished and thus it
is concluded that such finding is at least one of the main factors that elevate intrinsic activity. Computational analysis suggests, however, that
we should rethink this conclusion even if the aforementioned evidence indicates otherwise. If this conclusion is correct then the appropriate
homeostatic response that restores spontaneous activity should be the conservation of the pre-exposure population size for each ion channel type;
which elevates the number of channels per unit of membrane area for each distinct membrane conductance (see boxes below activity patterns in
1 and 2). The numerical recreation in 1 shows, nonetheless, that although this adaptation preserves RN and gK it did not prevent the elevation in
cell firing. Moreover, in 2 RN has dropped, while gK has increased yet the cell spiking is not decelerated but further augmented. Surprisingly, and
against the predominant descriptive reasoning that dominates the electrophysiological literature, 3 illustrates that the firing rate and spike waveform
are insensitive to changes in cell size as long as the number of channels per unit of membrane area is preserved. Altogether, it signifies that the
intrinsic firing pattern is only determined by the ion channel density of each conductance and not by the absolute magnitude of RN or gK. Notice that
such counterintuitive observation emerges as a consequence of computational analysis and not as a result of real experimentation. Consequently,
a hypothesis that experimental evidence seems to validate may be shown to be erroneous by numerical simulation.
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ERG channels population has dropped after size attenua-
tion. In agreement with this hypothesis, the addition of the
ERG channel into the DA cell model was able to effectively
buffer the rise in cell firing activity after size downscaling.
The geometry of the discharge pattern was, however, visu-
ally distinct to that of the control simulation; the spiking
pattern remains identical to the control state only when the
channel density of each conductance was clamped (Figure
4f). Consequently, to rescue the spike waveform appear-
ance, after size reduction, it is necessary to preserve the
temporal interrelations among all distinct ionic currents; it
cannot be achieved by changing the expression level of a
single conductance in the isolation of the others.
4.5
|
Cell size attenuation correlates with an
unusual improvement in synaptic integration
Synaptic summation in the model was paradoxically elevated
after size contraction since it consisted of a significantly re-
duced summation indexed but a substantially increased de-
polarization. This numerical finding agrees with our previous
experimental data that show that capacitance reduction im-
proves the temporal summation of synaptic inputs in exactly
the same manner as the simulations in this study (Arencibia-
Albite et al., 2017). This potentiated sub-threshold activity
was also associated with the enhancement of the burst signal
in response to a fixed train of synaptic currents; a result con-
sistent with data in mice showing that DA cell size reduction
increases burst spiking (Coque et al., 2011; Mazei-Robison
et al., 2011; Mcclung et al., 2005). These observations sug-
gest, overall, that the present model detail is adequate since
its behavior is in close agreement with relevant published
data.
5
|
CONCLUSION
Simulations in this study confirm that the experimentally
measured inverse relationship between gK and the rate of in-
trinsic activity is only valid when all other biophysical prop-
erties remain fairly unaltered or fixed. Reductions in gK that
happen in parallel to cell size contraction tend to depress
the augmentation in spontaneous activity. Computational
experiments illustrate that the appropriate homeostatic re-
sponse that conserves the pretreatment intrinsic spiking pat-
tern and the rate is the conservation of the channel density
of each conductance. This result is not a numerical artifact
as it was analytically demonstrated. The presence of the K+
ERG conductance is also effective in dampening the rise in
the autonomous firing as ERG channel activation increases
as the spike depolarization rate increases; however, the spik-
ing pattern remains clearly distinct from that of the control
condition. Additionally, with cell size reduction, the stand-
ard measurement of synaptic summation appears depress
yet the afferent-driven burst activity is improved due to the
enhanced depolarization secondary to the diminished cell
capacitance. On the whole, this work demonstrates that ex-
perimentation in the absence of mathematical models may
lead to the erroneous interpretation of the counterintuitive
aspects of empirical data (Figure 7).
ACKNOWLEDGMENTS
This work was supported by grants from the NIGMS
(2SC1GM084854-05A1) and NSF-PIRE (OISE-1545803) to
Carlos A. Jiménez-Rivera.
DISCLOSURE
No conflicts of interest, financial or otherwise, are declared
by the authors.
ORCID
Francisco Arencibia-Albite https://orcid.
org/0000-0001-8087-5998
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How to cite this article: Arencibia-Albite F, Jiménez-
Rivera CA. Computational and theoretical insights
into the homeostatic response to the decreased cell
size of midbrain dopamine neurons. Physiol Rep.
2021;00:e14709. https://doi.org/10.14814/phy2.14709
APPENDIX
How a diminished cell size, with no
change in the number of ion channels,
augments spontaneous activity?
An autonomous spiking cell is analogous to a capacitor
that is periodically charged and discharged. During each
spontaneous spike, for instance, the cell membrane is first
charged during the action potential upward phase and then
discharged during the downward phase. The rate (i.e., dVm/
dt) of such a process is independent of cell size and is ulti-
mately determined by ion channels densities. To appreciate
why, imagine, a 50pF membrane with only N=10,000 leak
channels each having a unitary conductance γ=1pS (input
conductance=Gm=Nγ=10,000pS) and a reversal poten-
tial E = −60 mV. If this membrane is current-clamped to
V0 = −20 mV and then release, we know from electricity
theory (Jack et al., 1975), that it would take about five time
constants (5τ=5Cm/Gm=5(50pF/10,000pS)=5(5ms)=
25ms) to reach the resting potential E; a result that is inde-
pendent of the value of V0. Notice also that τ is a function of
the leak channels density ρ = N/Sa since
Thus, if Cm is decreased by 30%, while the number of leak
channels (ch) is kept constant then ρ increases by 43%
causing τ to decrease by 1.5ms
Hence, if this 30% smaller membrane is released from
V0= −20mV then the time required to reach E is now de-
creased to 5τnew=5 (3.5ms)=17.5ms. As a result, in order
to conserve the pre-reduction average discharging rate (i.e.,
∆Vm/∆t=∆Vm/[5τ]=−40mV/25ms=−1.6mV/ms), the
number of leak channels has to drop by 30% as in doing so
the discharging time ∆t is preserved since channels density
remains unaltered
Furthermore, the preservation of channels density not only
conserves τ and the average discharging rate but also the in-
stantaneous rate dVm/dt. To understand this fact, observe that
the above current-clamp experiment obeys the following ini-
tial value problem:
Thus, the derivative of its solution is
where
umax
=𝛾
(
V
0
−E
)
is the maximal unitary current at the
start of the membrane discharge and so the product umaxρ is
the maximal current density. Equation (1) shows that the in-
stantaneous rate at which Vm decays from V0 is independent of
the size of the cell membrane and is only determined by ρ as
long as γ is unaltered. In other words, two passive cells of dif-
ferent sizes but with identical channels types and densities will
express the same instantaneous discharge rate when releasing
from V0. Therefore, as a claim in the third sentence of this para-
graph, the rate at which Vm evolves over time is independent of
cell size and is ultimately determined by ion channels density.
The above argument can be extended to any membrane
with active properties as follows: by analogy to Equation (1)
any spontaneously active membrane with n different con-
ductance types can be described by the following initial value
problem
𝜏
=Cm
Gm
=
Cm⋅Sa
N⋅𝛾=
Cm
(
N
Sa )
⋅𝛾
=
Cm
𝜌⋅𝛾
𝜌
old =
N
Saold
=
N
Cm∕
Cm
=
10, 000ch
5, 000𝜇m2=2ch∕𝜇m
2
→
𝜌new =10, 000ch
3, 500
𝜇
m2≈2.86ch∕𝜇m2
𝜏
new =
Cm
𝜌new ⋅𝛾=1 pF∕100 𝜇m2
(
2.86 ch∕𝜇m2
)
(1 pS∕ch)≈0.0035 sec .=
3.5 ms
𝜌
new =
7, 000 ch
3, 500 𝜇m2=2 ch∕𝜇m2=𝜌old
→
Δt=5𝜏new =5
Cm
𝜌
new
⋅𝛾=5
Cm
𝜌
old
⋅𝛾=5𝜏old =
25 ms
C
m⋅Sa
dV
m
dt
+N𝛾
(
Vm−E
)
=0, Vm(0)=V
0
(1)
d
Vm
dt=− 1
C
m
𝛾V0−E
N
Sa exp
−t
C
m
∕
N∕Sa
𝛾
=− 1
C
m
umax𝜌exp
−t
C
m
∕𝜌𝛾
(2)
d
Vm
dt=− 1
Cm
n
∑
j=1
𝜌j⋅uj
(
t,Vm
)
,Vm(0)=V
0
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where ρj is channel density for the j ion channel, while uj(t,Vm)
is its unitary current which is some empirically defined the
differentiable function of t and Vm. The solution to the problem
(2) defines the relationship between the rate of spontaneous
activity and channel densities. To see why to assume that from
t=0 to some other time t the membrane potential was con-
tinuously displaced from the initial condition Vm(0) = V0 to
Vm(t), that is,
Hence, the membrane potential at an arbitrary time t is
given by
Equation (3) states that in order to reproduce a par-
ticular firing behavior that starts from V0 we are
only required to know the set of channels densities
{
𝜌
j|
j:=1, 2, …,n
}
and the set of unitary current functions
{
u
j(
t,V
m
(t)
)|
j:=1, 2, …,n
}
; the size of the membrane is
unnecessary information. Consequently, two intrinsically
active membranes of distinct sizes, but with the same types
of ion channels, will show identical autonomous firing pat-
terns if and only if both membranes express the same set of
channel densities. Further validation of this conclusion is
given by a popular theorem in the field of differential equa-
tions: Theorem of the existence of a unique solution (Zill
et al., 2013). This theorem postulates that any initial value
problem of the form
manifests a unique solution if f(t,Vm) and ∂f/∂Vm are con-
tinuous functions in some region of the t-Vm plane
that contains the initial condition (t0, V0). In prob-
lem (2) f
t,Vm
=−
1
CmN
j=1𝜌j⋅uj
t,Vm
and
𝜕f
𝜕
V
m
=− 1
Cm∑
N
j=1𝜌j⋅
𝜕u
j
𝜕V
m
are both continuous functions since
each uj(t, Vm) is by definition a differentiable function. Hence,
Equation (3) is unique and so for a fixed set of unitary current
functions it can only be modified whenever the set of channel
densities is altered. Therefore, since the answer to the problem
(2) is the membrane's pattern of intrinsic activity the statement
of unique solution implies that, under cell size reduction, the
pretreatment spontaneous spiking behavior is only conserved
when the channel density of each conductance is preserved as
seen in our simulations.
V
m
(t)
∫
V
0
dVm
dtdt=− 1
Cm
t
∫
0
n
∑
j=1
𝜌j⋅uj
(
t,Vm(t)
)
d
t
(3)
V
m(t)=− 1
Cm
n
∑
j=1
𝜌j
t
∫
0
uj
(
t,Vm(t)
)
dt+V
0
dV
m
dt
=f
(
t,Vm
)
,Vm
(
t0
)
=V
0
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