1 Introduction
2 Methods
2.1 Cell morphology
S
total Total surface area |
S
soma Soma surface area |
S
axon Axon surface area |
S
dendrite Dendrite surface area |
D
axon Axon diameter |
D
dendrite Average dendrite diameter | |
---|---|---|---|---|---|---|
μm2
|
μm2
|
μm2
|
μm2
|
μm |
μm | |
Min | 969 | 10 | 17020 | 292 | 1 | 0.09 |
Max | 32457 | 2196 | 17020 | 31773 | 1 | 2.18 |
Ave | 9099 | 488 | 17020 | 8613 | 1 | 0.78 |
St Dev | 6004 | 425 | 0 | 6089 | 0 | 0.37 |
Number of dendritic tips |
R
dend,total Dendrite to total surface area ratio |
R
soma,total Soma to total surface area ratio | Number of bifurcations | Average dendritic compartment length μm | ||
---|---|---|---|---|---|---|
Min | 7 | 0.30 | 0.0005 | 13 | 8 | |
Max | 377 | 0.99 | 0.70 | 754 | 142 | |
Ave | 62 | 0.90 | 0.10 | 123 | 42 | |
St Dev | 44 | 0.14 | 0.14 | 88 | 26 |
2.2 Model parameters
Temperature | T=32 °C |
Membrane capacitance |
\(C_{m}=1 \; \mu \) F/cm2
|
Potassium reversal potential |
V
K = −70 mV |
Leak reversal potential |
V
L = −60 mV |
Hyperpolarization-activated reversal potential |
V
h = 0 mV |
LVA calcium reversal potential |
V
T = 120 mV |
Sodium reversal potential |
V
Na = 35 mV |
Calcium reversal potential |
V
Ca ≈ 120 mV at rest; is variable, see (2) |
Calcium dissociation constant | [Ca2+]diss = 10−6 M |
Extracellular calcium ion concentration | [Ca2+]e = 1.8 mM |
Gas constant |
R = 8.314 J/(M.K) |
Faraday constant |
F = 9.684 · 104 C/M |
Na+ channel |
\(\alpha _{m} = \frac {-0.6 (V+30)}{\mathrm {exp}(-0.1(V+30))-1}\)
|
\(\beta _{m} = 20 \mathrm {exp}\left (\frac {-(V+55)}{18}\right )\)
|
\(\alpha _{h} = 0.4 \mathrm {exp}\left (\frac {-(V+50)}{20}\right )\)
|
\(\beta _{h} = \frac {6}{1 + \mathrm {exp}(-0.1(V+20))}\)
| |
Ca2+ channel |
\(\alpha _{c} =\frac {-0.3(V+13)}{\mathrm {exp}(-0.1(V+13))-1}\)
|
\( \beta _{c} = 10 \mathrm {exp}\left (\frac {-(V + 38)}{18}\right )\)
|
K+ channel |
\(\alpha _{n} = \frac {-0.02 (V+40)}{\mathrm {exp}(-0.1(V+40))-1}\)
|
\(\beta _{n} = 0.4 \mathrm {exp}\left (\frac {-(V + 50)}{80}\right )\)
|
A channel |
\(\alpha _{a} = \frac {-0.006 (V+90)}{\mathrm {exp}(-0.1(V+90))-1}\)
|
\(\beta _{a} = 0.1 \mathrm {exp}\left (\frac {-(V + 30)}{10}\right )\)
|
\(\alpha _{h_{\mathrm {A}}} = 0.04 \mathrm {exp}\left (\frac {-(V+70)}{20}\right )\)
|
\(\beta _{h_{\mathrm {A}}} =\frac { 0.6}{1 + \mathrm {exp}(-0.1 (V+40))}\)
| |
h channel |
\(\alpha _{l} = \mathrm {exp}(0.08316(V+75))\)
|
\(\beta _l =\mathrm {exp}(0.033264(V+75))\)
|
T channel |
\(\alpha _{m_{\mathrm {T}}}= \left (1.7 + \mathrm {exp}\left ( \frac {-(V + 28.8)}{13.5} \right )\right )^{-1}\)
|
\(\beta _{m_{\mathrm {T}}}= \frac {1 + \mathrm {exp}({-(V +63)}{7.8})}{1.7 +\mathrm {exp}\left (\frac { -(V + 28.8)}{13.5}\right )}\)
|
\(\alpha _{h_{\mathrm {T}}} = \mathrm {exp}\left (\frac {-(V +160.3)}{17.8}\right )\)
|
\(\beta _{h_{T}}=\alpha _{h_{T}}\left [0.25 + \mathrm {exp}\left (\frac {V + 83.5}{6.3}\right )\right ]^{0.5}-0.5\alpha _{h_{T}}\)
| |
\(\alpha _{d} =\frac {1 + \mathrm {exp}\left (\frac {V + 37.4}{30}\right )}{240 \left [0.5 + \left (0.25 + \mathrm {exp}\left (\frac {V + 83.5}{6.3}\right )\right )^{0.5}\right ]} \)
|
\(\beta _{d} = \alpha _{d}\left [0.25 +\mathrm {exp}\left (\frac {V + 83.5}{6.3}\right )\right ]^{0.5}\)
| |
NaP channel | ||
[V<−40] |
\(\alpha _{p} = \frac {0.025 + 0.14 \mathrm {exp}\left (\frac { V + 40 }{ 10}\right )}{1 + \mathrm {exp}\left (\frac { - (V+48 ) }{ 10} \right )}\)
|
\(\beta _{p}=\frac { 1-\left (1 + \mathrm {exp}\left ( \frac { - (V+48 ) }{ 10} \right )\right )^{-1}}{0.025 + 0.14 \mathrm {exp}\left ( \frac { V + 40 }{ 10}\right )} \)
|
[V≥−40 |
\(\alpha _{p} = \frac {0.02 + 0.145 \mathrm {exp}\left ( \frac {-(V+ 40) }{ 10}\right )}{1 + \mathrm {exp}\left ( \frac { - (V+48)}{10} \right )}\)
|
\(\beta _{p}=\frac {1-\left (1 + \mathrm {exp}\left ( \frac { - (V+48 )}{10}\right )\right )^{-1}}{0.02 + 0.145 \mathrm {exp}\left ( \frac {-(V +40)}{10}\right )}\)
|
Soma | Dendrites | Initial segment | Narrow region | SOCB | Axon | |
---|---|---|---|---|---|---|
ḡ
Na
|
0.08
|
0.025
|
0.15
|
0.2
|
5 × 0.08
|
0.07
|
ḡ
Ca
|
0.0015
|
0.002
|
0.0015
| − | − | − |
ḡ
K
|
0.018
|
0.012
|
0.018
|
0.018
| − |
0.018
|
ḡ
K,A
|
0.054
|
0.036
|
0.054
| − | − | − |
ḡ
K(Ca)
|
0.000065
|
0.000001
|
0.000065
|
0.000065
| − | 0.07 |
ḡ
L
|
15 × \( {\bar{\mathbf{g}}}_{\mathbf{L_0}}\)
|
15 × \( {\bar{\mathbf{g}}}_{\mathbf{L}_0}\)
|
15 × \( {\bar{\mathbf{g}}}_{\mathbf{L}_0}\)
|
15 × \( {\bar{\mathbf{g}}}_{\mathbf{L}_0}\)
|
15 × \( {\bar{\mathbf{g}}}_{\mathbf{L}_0}\)
|
25 × \( {\bar{\mathbf{g}}}_{\mathrm{L}_0}\)
|
ḡ
h
|
ḡ
h
|
ḡ
h
|
ḡ
h
|
ḡ
h
| − |
ḡ
h
|
ḡ
NaP
|
ḡ
NaP
|
ḡ
NaP
| 0.05 × ḡ
NaP
| 0.05 × ḡ
NaP
| 5 × ḡ
NaP
| 0.05 × ḡ
NaP
|
ḡ
T
|
ḡ
T
| 5 × ḡ
T
|
ḡ
T
|
ḡ
T
|
ḡ
T
|
ḡ
T
|
2.3 Model tuning
-
The value of the mean resting potential for ON cells and the mean resting potential for OFF cells.
-
The frequency of the spontaneous activity in OFF sustained (OFF S) and OFF transient (OFF T) cells and the absence of spontaneous activity in ON cells.
-
The absence of action potentials during a hyperpolarizing step of −0.2 nA in both ON and OFF cells.
-
Presence (in OFF cells) or absence (in ON cells) of high frequency burst firing at the termination of a hyperpolarizing step of −0.2 nA.
Constraint | ON RGCs | Model Constraints | Experimental data |
---|---|---|---|
C0 | Spikes overshoot 0 mV | ||
C1 ON | resting potential | [−62, −70] mV | −65.5± 1.3 mV |
C2 ON | Spontaneous Frequency | 0 Hz | 0 Hz |
C3 ON | Spike rate during 500 ms stimulus of −0.2 nA | 0 Hz | 0 Hz |
C4 ON | Burst rate at the termination of 500 ms stimulus of −0.2 nA | 0 Hz | 0 Hz |
OFF RGCs | Model Constraints | Experimental data | |
C0 | Spikes overshoot 0 mV | ||
C1 OFF | resting potential | [−50, −62] mV | −54.6 ± 1.0 mV (OFF T) −55.1± 1.1 mV (OFF S) |
C2 OFF | Spontaneous Frequency | [15, 23] Hz (OFF T) | 20.3 ± 3.0 Hz (OFF T) |
[40, 48] Hz (OFF S) | 43.3 ± 3.5 Hz (OFF S) | ||
C3 OFF | Spike rate during 500 ms stimulus of −0.2 nA | 0 Hz | 0 Hz |
C4 OFF | Burst rate at the termination of 500 ms stimulus of −0.2 nA | Rebound excitation ≥ 2 × spontaneous frequency | Rebound excitation 60 Hz |
2.4 Model validation
-
V1: Subthreshold oscillations in OFF cells. When a hyperpolarizing current is applied, OFF cells should reveal oscillations in the subthreshold regime with a rate of 1–10 Hz, similar to the data of Margolis and Detwiler (2007).
-
V2: Disruption of spontaneous activity reveals bursting behavior in OFF cells. With negative current steps of increasing amplitude, OFF cells should exhibit a transition from regular spontaneous activity, to irregular activity, to burst firing, and then to silence, as shown experimentally by Margolis and Detwiler (2007).
-
V3: Coefficient of variation of the inter-spike interval in OFF cells. The coefficient of variation (CV) is defined as the standard deviation of the inter-spike interval divided by the mean. It is a common measure of the irregularity of spiking, in which a high value CV reflects high variability and a low value of CV reflects high regularity. The CV of the inter-spike interval (ISI) in OFF cells should increase with increasing level of hyperpolarization (Margolis and Detwiler 2007).
3 Results
3.1 Cell morphology
Cells meeting constraints | Cells not meeting constraints | |||
---|---|---|---|---|
Mean | Standard deviation | Mean | Standard deviation | |
R
dend,total
| 0.84 | 0.16 | 0.97 | 0.04 |
S
total (μm2) | 5230 | 2380 | 14226 | 5494 |
R
in (MΩ) | 172.8 | 88 | 68.7 | 25 |