Regulation of firing frequency in a computational model of a midbrain dopaminergic neuron

Abstract

Dopaminergic (DA) neurons of the mammalian midbrain exhibit unusually low firing frequencies in vitro. Furthermore, injection of depolarizing current induces depolarization block before high frequencies are achieved. The maximum steady and transient rates are about 10 and 20 Hz, respectively, despite the ability of these neurons to generate bursts at higher frequencies in vivo. We use a three-compartment model calibrated to reproduce DA neuron responses to several pharmacological manipulations to uncover mechanisms of frequency limitation. The model exhibits a slow oscillatory potential (SOP) dependent on the interplay between the L-type Ca²⁺ current and the small conductance K⁺ (SK) current that is unmasked by fast Na⁺ current block. Contrary to previous theoretical work, the SOP does not pace the steady spiking frequency in our model. The main currents that determine the spontaneous firing frequency are the subthreshold L-type Ca²⁺ and the A-type K⁺ currents. The model identifies the channel densities for the fast Na⁺ and the delayed rectifier K⁺ currents as critical parameters limiting the maximal steady frequency evoked by a depolarizing pulse. We hypothesize that the low maximal steady frequencies result from a low safety factor for action potential generation. In the model, the rate of Ca²⁺ accumulation in the distal dendrites controls the transient initial frequency in response to a depolarizing pulse. Similar results are obtained when the same model parameters are used in a multi-compartmental model with a realistic reconstructed morphology, indicating that the salient contributions of the dendritic architecture have been captured by the simpler model.

Appendix

In the following equations the subscript “i” indicates a nonspecific compartment, whereas the subscripts “d”, “p”, and “s” indicate distal dendritic compartment, proximal dendritic compartment and somatic compartment respectively. All compartments are considered cylindrical in shape with diameter and length given by d_i and L_i, respectively.

Equations governing membrane potential in each compartment:

$$ \begin{array}{*{20}{c}} { - {{\hbox{C}}_{\rm{m}}}\left( {{{{\hbox{d}}{{\hbox{V}}_{\rm{d}}}} \mathord{\left/{\vphantom {{{\hbox{d}}{{\hbox{V}}_{\rm{d}}}} {\hbox{dt}}}} \right.} {\hbox{dt}}}} \right) = {{\hbox{I}}_{{\rm{Na,d}}}} + {{\hbox{I}}_{{\rm{KA,d}}}} + {{\hbox{I}}_{{\rm{KV,d}}}} + {{\hbox{I}}_{{\rm{SK,d}}}} + {{\hbox{I}}_{{\rm{NaP,d}}}} + {{\hbox{I}}_{{\rm{Ca,L,d}}}} + {{\hbox{I}}_{{\rm{CaP,d}}}} + {{\hbox{I}}_{{\rm{L,d}}}} + {{\hbox{I}}_{\rm{dp}}}} \hfill \\{ - {{\hbox{C}}_{\rm{m}}}\left( {{{{\hbox{d}}{{\hbox{V}}_{\rm{p}}}} \mathord{\left/{\vphantom {{{\hbox{d}}{{\hbox{V}}_{\rm{p}}}} {\hbox{dt}}}} \right.} {\hbox{dt}}}} \right) = {{\hbox{I}}_{{\rm{Na,p}}}} + {{\hbox{I}}_{{\rm{KA,p}}}} + {{\hbox{I}}_{{\rm{KV,p}}}} + {{\hbox{I}}_{{\rm{SK,p}}}} + {{\hbox{I}}_{{\rm{NaP,p}}}} + {{\hbox{I}}_{{\rm{Ca,L,p}}}} + {{\hbox{I}}_{{\rm{CaP,p}}}} + {{\hbox{I}}_{{\rm{L,p}}}} + {{\hbox{I}}_{\rm{pd}}} + {{\hbox{I}}_{\rm{ps}}}} \hfill \\{ - {{\hbox{C}}_{\rm{m}}}\left( {{{{\hbox{d}}{{\hbox{V}}_{\rm{s}}}} \mathord{\left/{\vphantom {{{\hbox{d}}{{\hbox{V}}_{\rm{s}}}} {\hbox{dt}}}} \right.} {\hbox{dt}}}} \right) = {{\hbox{I}}_{{\rm{Na,s}}}} + {{\hbox{I}}_{{\rm{KA,s}}}} + {{\hbox{I}}_{{\rm{KV,s}}}} + {{\hbox{I}}_{{\rm{SK,s}}}} + {{\hbox{I}}_{{\rm{NaP,s}}}} + {{\hbox{I}}_{{\rm{Ca,L,s}}}} + {{\hbox{I}}_{{\rm{CaP,s}}}} + {{\hbox{I}}_{{\rm{L,s}}}} + {{\hbox{I}}_{\rm{sp}}} - {{\hbox{I}}_{\rm{stim}}}} \hfill \\\end{array} $$

Here I_stim is the stimulus current applied to the soma.

Linear leakage current:

$$ {{\hbox{I}}_{{\rm{L,i}}}} = {{\hbox{I}}_{{\rm{L,Na,i}}}} + {{\hbox{I}}_{{\rm{L}}{\rm{.K,i}}}} + {{\hbox{I}}_{{\rm{L,Ca,I}}}} $$

where

$$ \begin{array}{*{20}{c}} {{{\hbox{I}}_{{\rm{L,K,i}}}} = {{\hbox{g}}_{{\rm{L,K,i}}}}\left( {{{\hbox{V}}_{\rm{i}}}{ - }{{\hbox{E}}_{\rm{K}}}} \right);\,{{\hbox{I}}_{{\rm{L,Na,i}}}} = {{\hbox{g}}_{{\rm{L,Na,i}}}}\left( {{{\hbox{V}}_{\rm{i}}}{ - }{{\hbox{E}}_{{\rm{Na,i}}}}} \right);\,{{\hbox{I}}_{{\rm{L,Ca,i}}}} = {{\hbox{g}}_{{\rm{L,Ca,i}}}}\left( {{{\hbox{V}}_{\rm{i}}}{ - }{{\hbox{E}}_{\rm{ca}}}} \right);} \hfill \\{{{\hbox{E}}_{{\rm{Na,i}}}} = \left( {{{\hbox{RT}} \mathord{\left/{\vphantom {{\hbox{RT}} {\hbox{F}}}} \right.} {\hbox{F}}}} \right)\ln \left( {{{{{\left[ {\hbox{Na}} \right]}_{\rm{out}}}} \mathord{\left/{\vphantom {{{{\left[ {\hbox{Na}} \right]}_{\rm{out}}}} {{{\left[ {\hbox{Na}} \right]}_{{\rm{in,i}}}}}}} \right.} {{{\left[ {\hbox{Na}} \right]}_{{\rm{in,i}}}}}}} \right)} \hfill \\\end{array} $$

Sodium pump current:

$$ {{{I_{NaP,i}} = {I_{NaP,\max, i}}} \mathord{\left/{\vphantom {{{I_{NaP,i}} = {I_{NaP,\max, i}}} {\left( {1 + {{\left( {{{{{{K_{m,Na}}} \mathord{\left/{\vphantom {{{K_{m,Na}}} {\left[ {Na} \right]}}} \right.} {\left[ {Na} \right]}}}_{_{in,i}}}} \right)}^{1.5}}} \right)}}} \right.} {\left( {1 + {{\left( {{{{{{K_{m,Na}}} \mathord{\left/{\vphantom {{{K_{m,Na}}} {\left[ {Na} \right]}}} \right.} {\left[ {Na} \right]}}}_{_{in,i}}}} \right)}^{1.5}}} \right)}} $$

Sodium balance:

$$ \begin{array}{*{20}{c}} {{{{\hbox{d}}{{\left[ {\hbox{Na}} \right]}_{{\rm{in,d}}}}} \mathord{\left/{\vphantom {{{\hbox{d}}{{\left[ {\hbox{Na}} \right]}_{{\rm{in,d}}}}} {\hbox{dt}}}} \right.} {\hbox{dt}}} = {{4 \times {f_{\rm{d}}}\left( { - {{\hbox{I}}_{{\rm{Na,d}}}} - {{\hbox{I}}_{{\rm{L,Na,d}}}} - {3}{{\hbox{I}}_{{\rm{NaP,d}}}}} \right)} \mathord{\left/{\vphantom {{4 \times {f_{\rm{d}}}\left( { - {{\hbox{I}}_{{\rm{Na,d}}}} - {{\hbox{I}}_{{\rm{L,Na,d}}}} - {3}{{\hbox{I}}_{{\rm{NaP,d}}}}} \right)} {\left( {{{\hbox{d}}_{\rm{d}}}{\hbox{F}}} \right)}}} \right.} {\left( {{{\hbox{d}}_{\rm{d}}}{\hbox{F}}} \right)}}} \hfill \\{{{{\hbox{d}}{{\left[ {\hbox{Na}} \right]}_{{\rm{in,p}}}}} \mathord{\left/{\vphantom {{{\hbox{d}}{{\left[ {\hbox{Na}} \right]}_{{\rm{in,p}}}}} {{\hbox{dt}} = 4 \times {f_{\rm{p}}}{{\left( { - {{\hbox{I}}_{{\rm{Na,p}}}} - {{\hbox{I}}_{{\rm{L,Na,p}}}} - 3{{\hbox{I}}_{{\rm{NaP,p}}}}} \right)} \mathord{\left/{\vphantom {{\left( { - {{\hbox{I}}_{{\rm{Na,p}}}} - {{\hbox{I}}_{{\rm{L,Na,p}}}} - 3{{\hbox{I}}_{{\rm{NaP,p}}}}} \right)} {\left( {{{\hbox{d}}_{\rm{p}}}{\hbox{F}}} \right)}}} \right.} {\left( {{{\hbox{d}}_{\rm{p}}}{\hbox{F}}} \right)}}}}} \right.} {{\hbox{dt}} = 4 \times {f_{\rm{p}}}{{\left( { - {{\hbox{I}}_{{\rm{Na,p}}}} - {{\hbox{I}}_{{\rm{L,Na,p}}}} - 3{{\hbox{I}}_{{\rm{NaP,p}}}}} \right)} \mathord{\left/{\vphantom {{\left( { - {{\hbox{I}}_{{\rm{Na,p}}}} - {{\hbox{I}}_{{\rm{L,Na,p}}}} - 3{{\hbox{I}}_{{\rm{NaP,p}}}}} \right)} {\left( {{{\hbox{d}}_{\rm{p}}}{\hbox{F}}} \right)}}} \right.} {\left( {{{\hbox{d}}_{\rm{p}}}{\hbox{F}}} \right)}}}}} \hfill \\{{{{\hbox{d}}{{\left[ {\hbox{Na}} \right]}_{{\rm{in,s}}}}} \mathord{\left/{\vphantom {{{\hbox{d}}{{\left[ {\hbox{Na}} \right]}_{{\rm{in,s}}}}} {{\hbox{dt}} = 4 \times {f_{\rm{s}}}{{\left( { - {{\hbox{I}}_{{\rm{Na,s}}}} - {{\hbox{I}}_{{\rm{L,Na,s}}}} - 3{{\hbox{I}}_{{\rm{NaP,s}}}}} \right)} \mathord{\left/{\vphantom {{\left( { - {{\hbox{I}}_{{\rm{Na,s}}}} - {{\hbox{I}}_{{\rm{L,Na,s}}}} - 3{{\hbox{I}}_{{\rm{NaP,s}}}}} \right)} {\left( {{{\hbox{d}}_{\rm{s}}}{\hbox{F}}} \right)}}} \right.} {\left( {{{\hbox{d}}_{\rm{s}}}{\hbox{F}}} \right)}}}}} \right.} {{\hbox{dt}} = 4 \times {f_{\rm{s}}}{{\left( { - {{\hbox{I}}_{{\rm{Na,s}}}} - {{\hbox{I}}_{{\rm{L,Na,s}}}} - 3{{\hbox{I}}_{{\rm{NaP,s}}}}} \right)} \mathord{\left/{\vphantom {{\left( { - {{\hbox{I}}_{{\rm{Na,s}}}} - {{\hbox{I}}_{{\rm{L,Na,s}}}} - 3{{\hbox{I}}_{{\rm{NaP,s}}}}} \right)} {\left( {{{\hbox{d}}_{\rm{s}}}{\hbox{F}}} \right)}}} \right.} {\left( {{{\hbox{d}}_{\rm{s}}}{\hbox{F}}} \right)}}}}} \hfill \\\end{array} $$

Calcium pump current:

$$ {{{{\hbox{I}}_{{\rm{CaP,i}}}} = {{\hbox{I}}_{{\rm{CaP,max}}}}{{\left[ {{\hbox{C}}{{\hbox{a}}^{2 + }}} \right]}_{{\rm{in,i}}}}} \mathord{\left/{\vphantom {{{{\hbox{I}}_{{\rm{CaP,i}}}} = {{\hbox{I}}_{{\rm{CaP,max}}}}{{\left[ {{\hbox{C}}{{\hbox{a}}^{2 + }}} \right]}_{{\rm{in,i}}}}} {\left( {{{\left[ {{\hbox{C}}{{\hbox{a}}^{2 + }}} \right]}_{{\rm{in,i}}}} + {{\hbox{K}}_{{\rm{m,CaP}}}}} \right)}}} \right.} {\left( {{{\left[ {{\hbox{C}}{{\hbox{a}}^{2 + }}} \right]}_{{\rm{in,i}}}} + {{\hbox{K}}_{{\rm{m,CaP}}}}} \right)}} $$

Calcium Balance:

$$ {{{\hbox{d}}{{\left[ {{\hbox{C}}{{\hbox{a}}^{2 + }}} \right]}_{{\rm{in,i}}}}} \mathord{\left/{\vphantom {{{\hbox{d}}{{\left[ {{\hbox{C}}{{\hbox{a}}^{2 + }}} \right]}_{{\rm{in,i}}}}} {{\hbox{dt}} = 2 \times {f_{\rm{Ca}}}{{\left( {{{\hbox{I}}_{{\rm{Ca,L,i}}}} + {{\hbox{I}}_{{\rm{CaP,i}}}} + {{\hbox{I}}_{{\rm{L,Ca,i}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\hbox{I}}_{{\rm{Ca,L,i}}}} + {{\hbox{I}}_{{\rm{CaP,i}}}} + {{\hbox{I}}_{{\rm{L,Ca,i}}}}} \right)} {\left( {{{\hbox{d}}_{\rm{i}}}{\hbox{F}}} \right)}}} \right.} {\left( {{{\hbox{d}}_{\rm{i}}}{\hbox{F}}} \right)}}}}} \right.} {{\hbox{dt}} = 2 \times {f_{\rm{Ca}}}{{\left( {{{\hbox{I}}_{{\rm{Ca,L,i}}}} + {{\hbox{I}}_{{\rm{CaP,i}}}} + {{\hbox{I}}_{{\rm{L,Ca,i}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\hbox{I}}_{{\rm{Ca,L,i}}}} + {{\hbox{I}}_{{\rm{CaP,i}}}} + {{\hbox{I}}_{{\rm{L,Ca,i}}}}} \right)} {\left( {{{\hbox{d}}_{\rm{i}}}{\hbox{F}}} \right)}}} \right.} {\left( {{{\hbox{d}}_{\rm{i}}}{\hbox{F}}} \right)}}}} $$

Fast sodium current:

$$ \begin{array}{*{20}{c}} {{{\text{I}}_{{\text{Na,i}}}} = {{\text{g}}_{{\text{Na,i}}}}{{\text{m}}^3}_{\text{i}}{{\text{h}}_{\text{i}}}\left( {{{\text{V}}_{\text{i}}} - {{\text{E}}_{{\text{Na,i}}}}} \right)} \hfill \\ {{{{\text{d}}{{\text{m}}_{\text{i}}}} \mathord{\left/{\vphantom {{{\text{d}}{{\text{m}}_{\text{i}}}} {{\text{dt}}}}} \right.\kern-\nulldelimiterspace} {{\text{dt}}}} = {{\left[ {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{{\text{half,m,i}}}} - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{{\text{half,m,i}}}} - {{\text{V}}_{\text{i}}}} \right)} {6.0}}} \right.\kern-\nulldelimiterspace} {6.0}}} \right]} \right\} - {{\text{m}}_{\text{i}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{{\text{half,m,i}}}} - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{{\text{half,m,i}}}} - {{\text{V}}_{\text{i}}}} \right)} {6.0}}} \right.\kern-\nulldelimiterspace} {6.0}}} \right]} \right\} - {{\text{m}}_{\text{i}}}}}} \right]} \mathord{\left/{\vphantom {{\left[ {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{{\text{half,m,i}}}} - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{{\text{half,m,i}}}} - {{\text{V}}_{\text{i}}}} \right)} {6.0}}} \right.\kern-\nulldelimiterspace} {6.0}}} \right]} \right\} - {{\text{m}}_{\text{i}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{{\text{half,m,i}}}} - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{{\text{half,m,i}}}} - {{\text{V}}_{\text{i}}}} \right)} {6.0}}} \right.\kern-\nulldelimiterspace} {6.0}}} \right]} \right\} - {{\text{m}}_{\text{i}}}}}} \right]} {{\tau _{\text{m}}}}}} \right.\kern-\nulldelimiterspace} {{\tau _{\text{m}}}}}} \hfill \\ {{{{\text{d}}{{\text{h}}_{\text{i}}}} \mathord{\left/{\vphantom {{{\text{d}}{{\text{h}}_{\text{i}}}} {{\text{dt}} = {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{{\text{half,h,i}}}} - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{{\text{half,h,i}}}} - {{\text{V}}_{\text{i}}}} \right)} {7.8}}} \right.\kern-\nulldelimiterspace} {7.8}}} \right]} \right\} - {{\text{h}}_{\text{i}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{{\text{half,h,i}}}} - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{{\text{half,h,i}}}} - {{\text{V}}_{\text{i}}}} \right)} {7.8}}} \right.\kern-\nulldelimiterspace} {7.8}}} \right]} \right\} - {{\text{h}}_{\text{i}}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{{\text{half,h,i}}}} - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{{\text{half,h,i}}}} - {{\text{V}}_{\text{i}}}} \right)} {7.8}}} \right.\kern-\nulldelimiterspace} {7.8}}} \right]} \right\} - {{\text{h}}_{\text{i}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{{\text{half,h,i}}}} - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{{\text{half,h,i}}}} - {{\text{V}}_{\text{i}}}} \right)} {7.8}}} \right.\kern-\nulldelimiterspace} {7.8}}} \right]} \right\} - {{\text{h}}_{\text{i}}}}}} \right)} {{\tau _{\text{h}}}}}} \right.\kern-\nulldelimiterspace} {{\tau _{\text{h}}}}}}}} \right.\kern-\nulldelimiterspace} {{\text{dt}} = {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{{\text{half,h,i}}}} - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{{\text{half,h,i}}}} - {{\text{V}}_{\text{i}}}} \right)} {7.8}}} \right.\kern-\nulldelimiterspace} {7.8}}} \right]} \right\} - {{\text{h}}_{\text{i}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{{\text{half,h,i}}}} - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{{\text{half,h,i}}}} - {{\text{V}}_{\text{i}}}} \right)} {7.8}}} \right.\kern-\nulldelimiterspace} {7.8}}} \right]} \right\} - {{\text{h}}_{\text{i}}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{{\text{half,h,i}}}} - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{{\text{half,h,i}}}} - {{\text{V}}_{\text{i}}}} \right)} {7.8}}} \right.\kern-\nulldelimiterspace} {7.8}}} \right]} \right\} - {{\text{h}}_{\text{i}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{{\text{half,h,i}}}} - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{{\text{half,h,i}}}} - {{\text{V}}_{\text{i}}}} \right)} {7.8}}} \right.\kern-\nulldelimiterspace} {7.8}}} \right]} \right\} - {{\text{h}}_{\text{i}}}}}} \right)} {{\tau _{\text{h}}}}}} \right.\kern-\nulldelimiterspace} {{\tau _{\text{h}}}}}}}} \hfill \\ {{\tau _{\text{m}}} = {{1.0} \mathord{\left/{\vphantom {{1.0} {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{\text{i}}} + 45.0} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 45.0} \right)} {1.5}}} \right.\kern-\nulldelimiterspace} {1.5}}} \right]} \right\} - {{1.0} \mathord{\left/{\vphantom {{1.0} {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{\text{i}}} + 65} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 65} \right)} {0.5}}} \right.\kern-\nulldelimiterspace} {0.5}}} \right]} \right\} + 0.04}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{\text{i}}} + 65} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 65} \right)} {0.5}}} \right.\kern-\nulldelimiterspace} {0.5}}} \right]} \right\} + 0.04}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{\text{i}}} + 45.0} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 45.0} \right)} {1.5}}} \right.\kern-\nulldelimiterspace} {1.5}}} \right]} \right\} - {{1.0} \mathord{\left/{\vphantom {{1.0} {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{\text{i}}} + 65} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 65} \right)} {0.5}}} \right.\kern-\nulldelimiterspace} {0.5}}} \right]} \right\} + 0.04}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{\text{i}}} + 65} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 65} \right)} {0.5}}} \right.\kern-\nulldelimiterspace} {0.5}}} \right]} \right\} + 0.04}}}}} \hfill \\ {{\tau _{\text{h}}} = {{168.0} \mathord{\left/{\vphantom {{168.0} {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{\text{i}}} + 29} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 29} \right)} {4.5}}} \right.\kern-\nulldelimiterspace} {4.5}}} \right]} \right\} - {{168.0} \mathord{\left/{\vphantom {{168.0} {\left\{ {1 + \exp \left( {\left[ {{{\left( {{{\text{V}}_{\text{i}}} + 49} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 49} \right)} {2.0}}} \right.\kern-\nulldelimiterspace} {2.0}}} \right]} \right)} \right\} + 2.0}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left( {\left[ {{{\left( {{{\text{V}}_{\text{i}}} + 49} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 49} \right)} {2.0}}} \right.\kern-\nulldelimiterspace} {2.0}}} \right]} \right)} \right\} + 2.0}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{\text{i}}} + 29} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 29} \right)} {4.5}}} \right.\kern-\nulldelimiterspace} {4.5}}} \right]} \right\} - {{168.0} \mathord{\left/{\vphantom {{168.0} {\left\{ {1 + \exp \left( {\left[ {{{\left( {{{\text{V}}_{\text{i}}} + 49} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 49} \right)} {2.0}}} \right.\kern-\nulldelimiterspace} {2.0}}} \right]} \right)} \right\} + 2.0}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left( {\left[ {{{\left( {{{\text{V}}_{\text{i}}} + 49} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 49} \right)} {2.0}}} \right.\kern-\nulldelimiterspace} {2.0}}} \right]} \right)} \right\} + 2.0}}}}} \hfill \\ \end{array} $$

Calcium current:

$$ \begin{array}{*{20}{c}} {{{\text{I}}_{{\text{Ca,}}}}{\text{L,i}} = {{\text{g}}_{{\text{Ca,L}}}}{{\text{d}}_{{\text{L,i}}}}\left( {{{\text{V}}_{\text{i}}}{\text{ - }}{{\text{E}}_{{\text{Ca}}}}} \right)} \hfill \\ {{{{\text{d}}{{\text{d}}_{{\text{L,i}}}}} \mathord{\left/{\vphantom {{{\text{d}}{{\text{d}}_{{\text{L,i}}}}} {{\text{dt}} = {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ { - {{\left( {{{\text{V}}_i} + 45.0} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_i} + 45.0} \right)} {0.5}}} \right.\kern-\nulldelimiterspace} {0.5}}} \right]} \right\} - {{\text{d}}_{{\text{L,i}}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ { - {{\left( {{{\text{V}}_i} + 45.0} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_i} + 45.0} \right)} {0.5}}} \right.\kern-\nulldelimiterspace} {0.5}}} \right]} \right\} - {{\text{d}}_{{\text{L,i}}}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ { - {{\left( {{{\text{V}}_i} + 45.0} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_i} + 45.0} \right)} {0.5}}} \right.\kern-\nulldelimiterspace} {0.5}}} \right]} \right\} - {{\text{d}}_{{\text{L,i}}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ { - {{\left( {{{\text{V}}_i} + 45.0} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_i} + 45.0} \right)} {0.5}}} \right.\kern-\nulldelimiterspace} {0.5}}} \right]} \right\} - {{\text{d}}_{{\text{L,i}}}}}}} \right)} {{\tau _{{\text{dL}}}}}}} \right.\kern-\nulldelimiterspace} {{\tau _{{\text{dL}}}}}}}}} \right.\kern-\nulldelimiterspace} {{\text{dt}} = {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ { - {{\left( {{{\text{V}}_i} + 45.0} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_i} + 45.0} \right)} {0.5}}} \right.\kern-\nulldelimiterspace} {0.5}}} \right]} \right\} - {{\text{d}}_{{\text{L,i}}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ { - {{\left( {{{\text{V}}_i} + 45.0} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_i} + 45.0} \right)} {0.5}}} \right.\kern-\nulldelimiterspace} {0.5}}} \right]} \right\} - {{\text{d}}_{{\text{L,i}}}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ { - {{\left( {{{\text{V}}_i} + 45.0} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_i} + 45.0} \right)} {0.5}}} \right.\kern-\nulldelimiterspace} {0.5}}} \right]} \right\} - {{\text{d}}_{{\text{L,i}}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ { - {{\left( {{{\text{V}}_i} + 45.0} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_i} + 45.0} \right)} {0.5}}} \right.\kern-\nulldelimiterspace} {0.5}}} \right]} \right\} - {{\text{d}}_{{\text{L,i}}}}}}} \right)} {{\tau _{{\text{dL}}}}}}} \right.\kern-\nulldelimiterspace} {{\tau _{{\text{dL}}}}}}}}} \hfill \\ {{\tau _{{\text{dL}}}} = {\tau _{{\text{Ca}}}}\exp \left[ { - {{{{\left( {{{\text{V}}_{\text{i}}} + 70.0} \right)}^2}} \mathord{\left/{\vphantom {{{{\left( {{{\text{V}}_{\text{i}}} + 70.0} \right)}^2}} {625}}} \right.\kern-\nulldelimiterspace} {625}}} \right] + 0.3} \hfill \\ \end{array} $$

Delayed rectifier current:

$$ \begin{array}{*{20}{c}} {{{\text{I}}_{{\text{KV,i}}}} = {{\text{g}}_{{\text{KV,i}}}}{{\text{n}}^3}_{\text{i}}\left( {{{\text{V}}_{\text{i}}} - {{\text{E}}_{{\text{Ca}}}}} \right)} \hfill \\ {{{{\text{d}}{{\text{n}}_{\text{i}}}} \mathord{\left/{\vphantom {{{\text{d}}{{\text{n}}_{\text{i}}}} {{\text{dt}} = {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ {{{\left( { - 25.0 - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( { - 25.0 - {{\text{V}}_{\text{i}}}} \right)} {12.0}}} \right.\kern-\nulldelimiterspace} {12.0}}} \right]} \right\} - {{\text{n}}_{\text{i}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( { - 25.0 - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( { - 25.0 - {{\text{V}}_{\text{i}}}} \right)} {12.0}}} \right.\kern-\nulldelimiterspace} {12.0}}} \right]} \right\} - {{\text{n}}_{\text{i}}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ {{{\left( { - 25.0 - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( { - 25.0 - {{\text{V}}_{\text{i}}}} \right)} {12.0}}} \right.\kern-\nulldelimiterspace} {12.0}}} \right]} \right\} - {{\text{n}}_{\text{i}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( { - 25.0 - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( { - 25.0 - {{\text{V}}_{\text{i}}}} \right)} {12.0}}} \right.\kern-\nulldelimiterspace} {12.0}}} \right]} \right\} - {{\text{n}}_{\text{i}}}}}} \right)} {{\tau _{\text{n}}}}}} \right.\kern-\nulldelimiterspace} {{\tau _{\text{n}}}}}}}} \right.\kern-\nulldelimiterspace} {{\text{dt}} = {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ {{{\left( { - 25.0 - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( { - 25.0 - {{\text{V}}_{\text{i}}}} \right)} {12.0}}} \right.\kern-\nulldelimiterspace} {12.0}}} \right]} \right\} - {{\text{n}}_{\text{i}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( { - 25.0 - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( { - 25.0 - {{\text{V}}_{\text{i}}}} \right)} {12.0}}} \right.\kern-\nulldelimiterspace} {12.0}}} \right]} \right\} - {{\text{n}}_{\text{i}}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ {{{\left( { - 25.0 - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( { - 25.0 - {{\text{V}}_{\text{i}}}} \right)} {12.0}}} \right.\kern-\nulldelimiterspace} {12.0}}} \right]} \right\} - {{\text{n}}_{\text{i}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( { - 25.0 - {{\text{V}}_{\text{i}}}} \right)} \mathord{\left/{\vphantom {{\left( { - 25.0 - {{\text{V}}_{\text{i}}}} \right)} {12.0}}} \right.\kern-\nulldelimiterspace} {12.0}}} \right]} \right\} - {{\text{n}}_{\text{i}}}}}} \right)} {{\tau _{\text{n}}}}}} \right.\kern-\nulldelimiterspace} {{\tau _{\text{n}}}}}}}} \hfill \\ {{\tau _{\text{n}}} = {{{\tau _{{\text{Kv}}}}} \mathord{\left/{\vphantom {{{\tau _{{\text{Kv}}}}} {\left\{ {1 + \exp \left[ {{{\left( {{V_i} + 39.0} \right)} \mathord{\left/{\vphantom {{\left( {{V_i} + 39.0} \right)} {8.0}}} \right.\kern-\nulldelimiterspace} {8.0}}} \right]} \right\}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( {{V_i} + 39.0} \right)} \mathord{\left/{\vphantom {{\left( {{V_i} + 39.0} \right)} {8.0}}} \right.\kern-\nulldelimiterspace} {8.0}}} \right]} \right\}}} + 1.0} \hfill \\ \end{array} $$

Transient outward potassium current:

$$ \begin{array}{*{20}{c}} {{{\text{I}}_{{\text{KA,i}}}} = {{\text{g}}_{{\text{KA,i}}}}{\text{p}}_{\text{i}}^3{{\text{q}}_{\text{i}}}\left( {{{\text{V}}_{\text{i}}}{{\text{E}}_{\text{K}}}} \right)} \hfill \\ {{{{\text{d}}{{\text{q}}_{\text{i}}}} \mathord{\left/{\vphantom {{{\text{d}}{{\text{q}}_{\text{i}}}} {{\text{dt}} = {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{\text{i}}} + {{\text{V}}_{{\text{half,q,i}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + {{\text{V}}_{{\text{half,q,i}}}}} \right)} {{{\text{S}}_{{\text{q,i}}}}}}} \right.\kern-\nulldelimiterspace} {{{\text{S}}_{{\text{q,i}}}}}}} \right]} \right\} - {{\text{q}}_{\text{i}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{\text{i}}} + {{\text{V}}_{{\text{half,q,i}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + {{\text{V}}_{{\text{half,q,i}}}}} \right)} {{{\text{S}}_{{\text{q,i}}}}}}} \right.\kern-\nulldelimiterspace} {{{\text{S}}_{{\text{q,i}}}}}}} \right]} \right\} - {{\text{q}}_{\text{i}}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{\text{i}}} + {{\text{V}}_{{\text{half,q,i}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + {{\text{V}}_{{\text{half,q,i}}}}} \right)} {{{\text{S}}_{{\text{q,i}}}}}}} \right.\kern-\nulldelimiterspace} {{{\text{S}}_{{\text{q,i}}}}}}} \right]} \right\} - {{\text{q}}_{\text{i}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{\text{i}}} + {{\text{V}}_{{\text{half,q,i}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + {{\text{V}}_{{\text{half,q,i}}}}} \right)} {{{\text{S}}_{{\text{q,i}}}}}}} \right.\kern-\nulldelimiterspace} {{{\text{S}}_{{\text{q,i}}}}}}} \right]} \right\} - {{\text{q}}_{\text{i}}}}}} \right)} {20.0}}} \right.\kern-\nulldelimiterspace} {20.0}}}}} \right.\kern-\nulldelimiterspace} {{\text{dt}} = {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{\text{i}}} + {{\text{V}}_{{\text{half,q,i}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + {{\text{V}}_{{\text{half,q,i}}}}} \right)} {{{\text{S}}_{{\text{q,i}}}}}}} \right.\kern-\nulldelimiterspace} {{{\text{S}}_{{\text{q,i}}}}}}} \right]} \right\} - {{\text{q}}_{\text{i}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{\text{i}}} + {{\text{V}}_{{\text{half,q,i}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + {{\text{V}}_{{\text{half,q,i}}}}} \right)} {{{\text{S}}_{{\text{q,i}}}}}}} \right.\kern-\nulldelimiterspace} {{{\text{S}}_{{\text{q,i}}}}}}} \right]} \right\} - {{\text{q}}_{\text{i}}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{\text{i}}} + {{\text{V}}_{{\text{half,q,i}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + {{\text{V}}_{{\text{half,q,i}}}}} \right)} {{{\text{S}}_{{\text{q,i}}}}}}} \right.\kern-\nulldelimiterspace} {{{\text{S}}_{{\text{q,i}}}}}}} \right]} \right\} - {{\text{q}}_{\text{i}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ {{{\left( {{{\text{V}}_{\text{i}}} + {{\text{V}}_{{\text{half,q,i}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + {{\text{V}}_{{\text{half,q,i}}}}} \right)} {{{\text{S}}_{{\text{q,i}}}}}}} \right.\kern-\nulldelimiterspace} {{{\text{S}}_{{\text{q,i}}}}}}} \right]} \right\} - {{\text{q}}_{\text{i}}}}}} \right)} {20.0}}} \right.\kern-\nulldelimiterspace} {20.0}}}}} \hfill \\ {{{{\text{d}}{{\text{p}}_{\text{i}}}} \mathord{\left/{\vphantom {{{\text{d}}{{\text{p}}_{\text{i}}}} {{\text{dt}} = {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ { - {{\left( {{{\text{V}}_{\text{i}}} + 43.0} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 43.0} \right)} {24.0}}} \right.\kern-\nulldelimiterspace} {24.0}}} \right]} \right\} - {{\text{p}}_{\text{i}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ { - {{\left( {{{\text{V}}_{\text{i}}} + 43.0} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 43.0} \right)} {24.0}}} \right.\kern-\nulldelimiterspace} {24.0}}} \right]} \right\} - {{\text{p}}_{\text{i}}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ { - {{\left( {{{\text{V}}_{\text{i}}} + 43.0} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 43.0} \right)} {24.0}}} \right.\kern-\nulldelimiterspace} {24.0}}} \right]} \right\} - {{\text{p}}_{\text{i}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ { - {{\left( {{{\text{V}}_{\text{i}}} + 43.0} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 43.0} \right)} {24.0}}} \right.\kern-\nulldelimiterspace} {24.0}}} \right]} \right\} - {{\text{p}}_{\text{i}}}}}} \right)} {{\tau _{\text{p}}}}}} \right.\kern-\nulldelimiterspace} {{\tau _{\text{p}}}}}}}} \right.\kern-\nulldelimiterspace} {{\text{dt}} = {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ { - {{\left( {{{\text{V}}_{\text{i}}} + 43.0} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 43.0} \right)} {24.0}}} \right.\kern-\nulldelimiterspace} {24.0}}} \right]} \right\} - {{\text{p}}_{\text{i}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ { - {{\left( {{{\text{V}}_{\text{i}}} + 43.0} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 43.0} \right)} {24.0}}} \right.\kern-\nulldelimiterspace} {24.0}}} \right]} \right\} - {{\text{p}}_{\text{i}}}}}} \right)} \mathord{\left/{\vphantom {{\left( {{1 \mathord{\left/{\vphantom {1 {\left\{ {1 + \exp \left[ { - {{\left( {{{\text{V}}_{\text{i}}} + 43.0} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 43.0} \right)} {24.0}}} \right.\kern-\nulldelimiterspace} {24.0}}} \right]} \right\} - {{\text{p}}_{\text{i}}}}}} \right.\kern-\nulldelimiterspace} {\left\{ {1 + \exp \left[ { - {{\left( {{{\text{V}}_{\text{i}}} + 43.0} \right)} \mathord{\left/{\vphantom {{\left( {{{\text{V}}_{\text{i}}} + 43.0} \right)} {24.0}}} \right.\kern-\nulldelimiterspace} {24.0}}} \right]} \right\} - {{\text{p}}_{\text{i}}}}}} \right)} {{\tau _{\text{p}}}}}} \right.\kern-\nulldelimiterspace} {{\tau _{\text{p}}}}}}}} \hfill \\ {{\tau _{\text{p}}} = 2.0\,\exp \left[ { - {{{{\left( {{{\text{V}}_i} + 50.0} \right)}^2}} \mathord{\left/{\vphantom {{{{\left( {{{\text{V}}_i} + 50.0} \right)}^2}} {550.0}}} \right.\kern-\nulldelimiterspace} {550.0}}} \right] + 1.1} \hfill \\ \end{array} $$

SK potassium current:

$$ {{\text{I}}_{{\text{SK,i}}}} = {{{{\text{g}}_{{\text{SK}}}}} \mathord{\left/{\vphantom {{{{\text{g}}_{{\text{SK}}}}} {\left[ {1 + {{\left( {{{{{\text{K}}_{{\text{m,SK}}}}} \mathord{\left/{\vphantom {{{{\text{K}}_{{\text{m,SK}}}}} {{{\left[ {{\text{C}}{{\text{a}}^{2 + }}} \right]}_{{\text{in,i}}}}}}} \right.\kern-\nulldelimiterspace} {{{\left[ {{\text{C}}{{\text{a}}^{2 + }}} \right]}_{{\text{in,i}}}}}}} \right)}^4}} \right]\left( {{{\text{V}}_{\text{i}}} - {{\text{E}}_{\text{K}}}} \right)}}} \right.\kern-\nulldelimiterspace} {\left[ {1 + {{\left( {{{{{\text{K}}_{{\text{m,SK}}}}} \mathord{\left/{\vphantom {{{{\text{K}}_{{\text{m,SK}}}}} {{{\left[ {{\text{C}}{{\text{a}}^{2 + }}} \right]}_{{\text{in,i}}}}}}} \right.\kern-\nulldelimiterspace} {{{\left[ {{\text{C}}{{\text{a}}^{2 + }}} \right]}_{{\text{in,i}}}}}}} \right)}^4}} \right]\left( {{{\text{V}}_{\text{i}}} - {{\text{E}}_{\text{K}}}} \right)}} $$

Intercompartmental coupling currents:

$$ \begin{array}{*{20}{c}} {{{\text{I}}_{{\text{dp}}}} = {{\text{g}}_{{\text{dp}}}}\left( {{{\text{V}}_{\text{d}}} - {{\text{V}}_{\text{p}}}} \right);\,{{\text{I}}_{{\text{pd}}}} = {{\text{g}}_{{\text{pd}}}}\left( {{{\text{V}}_{\text{p}}} - {{\text{V}}_{\text{d}}}} \right);\,{{\text{I}}_{{\text{ps}}}} = {{\text{g}}_{{\text{ps}}}}\left( {{{\text{V}}_{\text{p}}} - {{\text{V}}_s}} \right);{{\text{I}}_{{\text{sp}}}} = {{\text{g}}_{{\text{sp}}}}\left( {{{\text{V}}_s} - {{\text{V}}_{\text{p}}}} \right);} \hfill \\ {{{\text{g}}_{{\text{sp}}}} = {{4 \times {{10}^8}{{\text{G}}_{{\text{sp}}}}} \mathord{\left/{\vphantom {{4 \times {{10}^8}{{\text{G}}_{{\text{sp}}}}} {\left( {\pi {{\text{d}}_{\text{p}}}{{\text{L}}_{\text{p}}}} \right);{{\text{g}}_{{\text{ps}}}} = {{{{10}^8}{{\text{G}}_{{\text{sp}}}}} \mathord{\left/{\vphantom {{{{10}^8}{{\text{G}}_{{\text{sp}}}}} {\left( {\pi {{\text{d}}_{\text{p}}}{{\text{L}}_{\text{p}}}} \right);}}} \right.\kern-\nulldelimiterspace} {\left( {\pi {{\text{d}}_{\text{p}}}{{\text{L}}_{\text{p}}}} \right);}}}}} \right.\kern-\nulldelimiterspace} {\left( {\pi {{\text{d}}_{\text{p}}}{{\text{L}}_{\text{p}}}} \right);{{\text{g}}_{{\text{ps}}}} = {{{{10}^8}{{\text{G}}_{{\text{sp}}}}} \mathord{\left/{\vphantom {{{{10}^8}{{\text{G}}_{{\text{sp}}}}} {\left( {\pi {{\text{d}}_{\text{p}}}{{\text{L}}_{\text{p}}}} \right);}}} \right.\kern-\nulldelimiterspace} {\left( {\pi {{\text{d}}_{\text{p}}}{{\text{L}}_{\text{p}}}} \right);}}}}} \hfill \\ {{{\text{g}}_{{\text{pd}}}} = {{2 \times {{10}^8}{{\text{G}}_{{\text{pd}}}}} \mathord{\left/{\vphantom {{2 \times {{10}^8}{{\text{G}}_{{\text{pd}}}}} {\left( {\pi {{\text{d}}_{\text{p}}}{{\text{L}}_{\text{p}}}} \right);{{\text{g}}_{{\text{dp}}}} = {{{{10}^8}{{\text{G}}_{{\text{pd}}}}} \mathord{\left/{\vphantom {{{{10}^8}{{\text{G}}_{{\text{pd}}}}} {\left( {\pi {{\text{d}}_{\text{d}}}{{\text{L}}_{\text{d}}}} \right);}}} \right.\kern-\nulldelimiterspace} {\left( {\pi {{\text{d}}_{\text{d}}}{{\text{L}}_{\text{d}}}} \right);}}}}} \right.\kern-\nulldelimiterspace} {\left( {\pi {{\text{d}}_{\text{p}}}{{\text{L}}_{\text{p}}}} \right);{{\text{g}}_{{\text{dp}}}} = {{{{10}^8}{{\text{G}}_{{\text{pd}}}}} \mathord{\left/{\vphantom {{{{10}^8}{{\text{G}}_{{\text{pd}}}}} {\left( {\pi {{\text{d}}_{\text{d}}}{{\text{L}}_{\text{d}}}} \right);}}} \right.\kern-\nulldelimiterspace} {\left( {\pi {{\text{d}}_{\text{d}}}{{\text{L}}_{\text{d}}}} \right);}}}}} \hfill \\ {{{\text{G}}_{{\text{sp}}}} = {{{{10}^2}\pi {\text{d}}_{\text{p}}^{\text{2}}{\text{d}}_{\text{s}}^{\text{2}}} \mathord{\left/{\vphantom {{{{10}^2}\pi {\text{d}}_{\text{p}}^{\text{2}}{\text{d}}_{\text{s}}^{\text{2}}} {\left[ {2{{\text{R}}_{\text{a}}}\left( {{{\text{L}}_{\text{p}}}{\text{d}}_{\text{s}}^{\text{2}} + {{\text{L}}_{\text{s}}}{\text{d}}_{\text{p}}^{\text{2}}} \right)} \right]}}} \right.\kern-\nulldelimiterspace} {\left[ {2{{\text{R}}_{\text{a}}}\left( {{{\text{L}}_{\text{p}}}{\text{d}}_{\text{s}}^{\text{2}} + {{\text{L}}_{\text{s}}}{\text{d}}_{\text{p}}^{\text{2}}} \right)} \right]}}} \hfill \\ {{{\text{G}}_{{\text{pd}}}} = {{{{10}^2}\pi {\text{d}}_{\text{p}}^{\text{2}}{\text{d}}_{\text{d}}^{\text{2}}} \mathord{\left/{\vphantom {{{{10}^2}\pi {\text{d}}_{\text{p}}^{\text{2}}{\text{d}}_{\text{d}}^{\text{2}}} {\left[ {2{{\text{R}}_{\text{a}}}\left( {{{\text{L}}_{\text{p}}}{\text{d}}_{\text{d}}^2 + {{\text{L}}_{\text{d}}}{\text{d}}_{\text{p}}^2} \right)} \right]}}} \right.\kern-\nulldelimiterspace} {\left[ {2{{\text{R}}_{\text{a}}}\left( {{{\text{L}}_{\text{p}}}{\text{d}}_{\text{d}}^2 + {{\text{L}}_{\text{d}}}{\text{d}}_{\text{p}}^2} \right)} \right]}}} \hfill \\ \end{array} $$

Parameters

The same parameters were used for the models with schematic morphology and reconstructed morphology. The only difference was in the morphology.

$$ \begin{array}{*{20}{c}} {{{\text{C}}_{\text{m}}} = {{1\mu {\text{F}}} \mathord{\left/{\vphantom {{1\mu {\text{F}}} {{\text{c}}{{\text{m}}^2};{{\text{R}}_{\text{a}}} = 40\,\Omega \,{\text{cm}};{\text{R}} = {{8,314\,{\text{J}}} \mathord{\left/{\vphantom {{8,314\,{\text{J}}} {{\text{kg}}}}} \right.\kern-\nulldelimiterspace} {{\text{kg}}}}\,{\text{mol}}\,{\text{K}};{\text{F}} = {{96520\,{\text{C}}} \mathord{\left/{\vphantom {{96520\,{\text{C}}} {{\text{mol}};\,{\text{T}} = 308.15\,{\text{K}};}}} \right.\kern-\nulldelimiterspace} {{\text{mol}};\,{\text{T}} = 308.15\,{\text{K}};}}}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};{{\text{R}}_{\text{a}}} = 40\,\Omega \,{\text{cm}};{\text{R}} = {{8,314\,{\text{J}}} \mathord{\left/{\vphantom {{8,314\,{\text{J}}} {{\text{kg}}}}} \right.\kern-\nulldelimiterspace} {{\text{kg}}}}\,{\text{mol}}\,{\text{K}};{\text{F}} = {{96520\,{\text{C}}} \mathord{\left/{\vphantom {{96520\,{\text{C}}} {{\text{mol}};\,{\text{T}} = 308.15\,{\text{K}};}}} \right.\kern-\nulldelimiterspace} {{\text{mol}};\,{\text{T}} = 308.15\,{\text{K}};}}}}} \hfill \\ {{{\text{d}}_{\text{d}}} = 1.5\mu {\text{m}};\,{d_p} = 3\mu {\text{m}};{{\text{d}}_{\text{s}}} = 15\mu {\text{m}};{{\text{L}}_{\text{p}}} = 150\mu {\text{m}};{{\text{L}}_{\text{s}}} = 25\mu {\text{m}};} \hfill \\ {{{\text{E}}_{\text{k}}} = - 90\,{\text{mV}};{{\text{E}}_{{\text{Ca}}}} = 120{\text{mV}};{{\text{V}}_{{\text{half,m,i}}}} = - 34.6\,{\text{mV}};{{\text{V}}_{{\text{half,h,i}}}} = - 56.8\,{\text{mV}};} \hfill \\ {{\tau _{{{\text{K}}_{\text{v}}}}} = 19.0\,{\text{ms}};{\tau _{{\text{Ca}}}} = 18.0\,{\text{ms}};} \hfill \\ {{{\left[ {{\text{Na}}} \right]}_{{\text{out}}}} = 145\,{\text{mM}};{{\text{K}}_{{\text{M,SK}}}} = 0.00019\,{\text{mM}};{{\text{K}}_{{\text{M,CaP}}}} = 0.0005\,{\text{M}} = {\text{mM}};{{\text{K}}_{{\text{M,Na}}}} = 10\,{\text{mM}};} \hfill \\ {{f_i} = 2;\,{{\text{F}}_{{\text{Ca}}}} = 0.05;} \hfill \\ {{{\text{I}}_{{\text{CaP,max,i}}}} = {{0.00191\,{\text{mA}}} \mathord{\left/{\vphantom {{0.00191\,{\text{mA}}} {{\text{c}}{{\text{m}}^2};{{\text{I}}_{{\text{NaP,max,i}}}} = {{0.0092\,{\text{mA}}} \mathord{\left/{\vphantom {{0.0092\,{\text{mA}}} {{\text{c}}{{\text{m}}^2};}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};}}}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};{{\text{I}}_{{\text{NaP,max,i}}}} = {{0.0092\,{\text{mA}}} \mathord{\left/{\vphantom {{0.0092\,{\text{mA}}} {{\text{c}}{{\text{m}}^2};}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};}}}}} \hfill \\ {{{\text{g}}_{{\text{L,Na,i}}}} = {{2.375\,\mu {\text{S}}} \mathord{\left/{\vphantom {{2.375\,\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};{{\text{g}}_{{\text{L,Ca,i}}}} = {{0.0136\,\mu {\text{S}}} \mathord{\left/{\vphantom {{0.0136\,\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};{{\text{g}}_{{\text{L,K,i}}}} = {{5.5\,\mu {\text{S}}} \mathord{\left/{\vphantom {{5.5\,\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};}}}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};{{\text{g}}_{{\text{L,K,i}}}} = {{5.5\,\mu {\text{S}}} \mathord{\left/{\vphantom {{5.5\,\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};}}}}}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};{{\text{g}}_{{\text{L,Ca,i}}}} = {{0.0136\,\mu {\text{S}}} \mathord{\left/{\vphantom {{0.0136\,\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};{{\text{g}}_{{\text{L,K,i}}}} = {{5.5\,\mu {\text{S}}} \mathord{\left/{\vphantom {{5.5\,\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};}}}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};{{\text{g}}_{{\text{L,K,i}}}} = {{5.5\,\mu {\text{S}}} \mathord{\left/{\vphantom {{5.5\,\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};}}}}}}} \hfill \\ {{{\text{g}}_{{\text{Na,i}}}} = {{550\,\mu {\text{S}}} \mathord{\left/{\vphantom {{550\,\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};{{\text{g}}_{{\text{KV,i}}}} = {{665\,\mu {\text{S}}} \mathord{\left/{\vphantom {{665\,\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};}}}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};{{\text{g}}_{{\text{KV,i}}}} = {{665\,\mu {\text{S}}} \mathord{\left/{\vphantom {{665\,\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};}}}}} \hfill \\ {{{\text{g}}_{{\text{Ca,L,i}}}} = {{11.196\,\mu {\text{S}}} \mathord{\left/{\vphantom {{11.196\,\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};{{\text{g}}_{{\text{SK,i}}}} = {{59\,\mu {\text{S}}} \mathord{\left/{\vphantom {{59\,\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};}}}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};{{\text{g}}_{{\text{SK,i}}}} = {{59\,\mu {\text{S}}} \mathord{\left/{\vphantom {{59\,\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};}}}}} \hfill \\ {{{\text{g}}_{{\text{KA,d}}}} = {{266\,\mu {\text{S}}} \mathord{\left/{\vphantom {{266\,\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};{{\text{g}}_{{\text{KA,p}}}} = {{285\,\mu {\text{S}}} \mathord{\left/{\vphantom {{285\,\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};{{\text{g}}_{{\text{KA,s}}}} = 570\,{{\mu {\text{S}}} \mathord{\left/{\vphantom {{\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};}}}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};{{\text{g}}_{{\text{KA,s}}}} = 570\,{{\mu {\text{S}}} \mathord{\left/{\vphantom {{\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};}}}}}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};{{\text{g}}_{{\text{KA,p}}}} = {{285\,\mu {\text{S}}} \mathord{\left/{\vphantom {{285\,\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};{{\text{g}}_{{\text{KA,s}}}} = 570\,{{\mu {\text{S}}} \mathord{\left/{\vphantom {{\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};}}}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};{{\text{g}}_{{\text{KA,s}}}} = 570\,{{\mu {\text{S}}} \mathord{\left/{\vphantom {{\mu {\text{S}}} {{\text{c}}{{\text{m}}^2};}}} \right.\kern-\nulldelimiterspace} {{\text{c}}{{\text{m}}^2};}}}}}}} \hfill \\ \end{array} $$