Somato-dendritic mechanisms underlying the electrophysiological properties of hypothalamic magnocellular neuroendocrine cells: A multicompartmental model study

Abstract

Magnocellular neuroendocrine cells (MNCs) of the hypothalamus synthesize the neurohormones vasopressin and oxytocin, which are released into the blood and exert a wide spectrum of actions, including the regulation of cardiovascular and reproductive functions. Vasopressin- and oxytocin-secreting neurons have similar morphological structure and electrophysiological characteristics. A realistic multicompartmental model of a MNC with a bipolar branching structure was developed and calibrated based on morphological and in vitro electrophysiological data in order to explore the roles of ion currents and intracellular calcium dynamics in the intrinsic electrical MNC properties. The model was used to determine the likely distributions of ion conductances in morphologically distinct parts of the MNCs: soma, primary dendrites and secondary dendrites. While reproducing the general electrophysiological features of MNCs, the model demonstrates that the differential spatial distributions of ion channels influence the functional expression of MNC properties, and reveals the potential importance of dendritic conductances in these properties.

Appendices

Appendix

The membrane potential of each compartment of the magnocellular neuron is described by the current balance equation:

$$ \begin{array}{*{20}c} {C_{m} {\left( {{{\text{d}}V_{{{\text{sd}}}} } \mathord{\left/ {\vphantom {{{\text{d}}V_{{{\text{sd}}}} } {{\text{d}}t}}} \right. \kern-\nulldelimiterspace} {{\text{d}}t}} \right)} = I_{{{\text{Na,sd}}}} + I_{{A,{\text{sd}}}} + I_{{K{\left( {DR} \right)},{\text{sd}}}} + I_{{{\text{Ca}},L,{\text{sd}}}} + I_{{{\text{CAN,sd}}}} + I_{{{\text{SK,sd}}}} + I_{{L,{\text{sd}}}} + I_{{{\text{sd}} - pd}} } \\ {C_{m} {\left( {{{\text{d}}V_{{{\text{pd}}}} } \mathord{\left/ {\vphantom {{{\text{d}}V_{{{\text{pd}}}} } {{\text{d}}t}}} \right. \kern-\nulldelimiterspace} {{\text{d}}t}} \right)} = I_{{{\text{Na}},{\text{pd}}}} + I_{{A,{\text{pd}}}} + I_{{K{\left( {DR} \right)},{\text{pd}}}} + I_{{{\text{Ca}},L,{\text{pd}}}} + I_{{{\text{Ca}},N,{\text{pd}}}} + I_{{{\text{SK}},{\text{pd}}}} + I_{{{\text{BK}},{\text{pd}}}} + I_{{{\text{CAN}},{\text{pd}}}} + I_{{L,{\text{pd}}}} + I_{{{\text{pd - s}}}} + I_{{{\text{pd - sd}}}} } \\ {C_{m} {\left( {{{\text{d}}V_{{\text{s}}} } \mathord{\left/ {\vphantom {{{\text{d}}V_{{\text{s}}} } {{\text{d}}t}}} \right. \kern-\nulldelimiterspace} {{\text{d}}t}} \right)} = I_{{{\text{Na}},{\text{s}}}} + I_{{A,{\text{s}}}} + I_{{K{\left( {DR} \right)},{\text{s}}}} + I_{{{\text{Ca}},L,{\text{s}}}} + I_{{{\text{Ca}},N,{\text{s}}}} + I_{{{\text{SK}},{\text{s}}}} + I_{{{\text{BK}},{\text{s}}}} + I_{{{\text{CAN}},{\text{s}}}} + I_{{L,{\text{s}}}} + I_{{{\text{s - pd}}}} + I_{{{\text{stim}}}} ,} \\ \end{array} $$

where C _m is the specific neuronal membrane capacitance; I _sd-pd, I _pd-s , I _pd-sd, and I _s-pd are the intercompartmental coupling currents; and I _stim is the applied microelectrode current. The subscripts “sd,” “pd,” and “s” indicate secondary (distal) dendritic compartment, primary (proximal) dendritic compartment, and somatic compartment, respectively. Each compartment has its own set of conductances for the ion currents.

Membrane ion currents

The membrane ion currents in the model were described using the Hodgkin-Huxley formalism. The subscript “i” indicates a compartment (s, pd or sd).

2.1 Transient sodium current (based on Tanaka et al. 1999)

$$ \begin{array}{*{20}l} {{I_{{{\text{Na}},i}} = g_{{{\text{Na}},i}} m^{3}_{\infty } ih_{i} {\left( {V_{i} - E_{{{\text{Na}}}} } \right)};} \hfill} \\ {{m_{{\infty ,i}} = 1 \mathord{\left/ {\vphantom {1 {{\left( {1 + \exp {\left( { - {{\left( {V_{i} + 34.6} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} + 34.6} \right)}} {6.2}}} \right. \kern-\nulldelimiterspace} {6.2}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left( { - {{\left( {V_{i} + 34.6} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} + 34.6} \right)}} {6.2}}} \right. \kern-\nulldelimiterspace} {6.2}} \right)}} \right)}};} \hfill} \\ {{{{\text{d}}h_{i} } \mathord{\left/ {\vphantom {{{\text{d}}h_{i} } {{\text{d}}t}}} \right. \kern-\nulldelimiterspace} {{\text{d}}t} = {{\left( {h_{{\infty ,i}} - h_{i} } \right)}} \mathord{\left/ {\vphantom {{{\left( {h_{{\infty ,i}} - h_{i} } \right)}} {\tau _{{h,i}} }}} \right. \kern-\nulldelimiterspace} {\tau _{{h,i}} };} \hfill} \\ {{h_{{\infty ,i}} = 1 \mathord{\left/ {\vphantom {1 {{\left( {1 + \exp {\left( {{{\left( {V_{i} + 61.6} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} + 61.6} \right)}} {6.8}}} \right. \kern-\nulldelimiterspace} {6.8}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left( {{{\left( {V_{i} + 61.6} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} + 61.6} \right)}} {6.8}}} \right. \kern-\nulldelimiterspace} {6.8}} \right)}} \right)}};} \hfill} \\ {{\tau _{{h,i}} = {26.0} \mathord{\left/ {\vphantom {{26.0} {{\left( {1 + \exp {\left( {{{\left( {V_{i} + 46.0} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} + 46.0} \right)}} {7.0}}} \right. \kern-\nulldelimiterspace} {7.0}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left( {{{\left( {V_{i} + 46.0} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} + 46.0} \right)}} {7.0}}} \right. \kern-\nulldelimiterspace} {7.0}} \right)}} \right)}} + {3.0} \mathord{\left/ {\vphantom {{3.0} {{\left( {1 + \exp {\left( {{{\left( {V_{i} + 49.0} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} + 49.0} \right)}} {20.0}}} \right. \kern-\nulldelimiterspace} {20.0}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left( {{{\left( {V_{i} + 49.0} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} + 49.0} \right)}} {20.0}}} \right. \kern-\nulldelimiterspace} {20.0}} \right)}} \right)}} + 0.1.} \hfill} \\ \end{array} $$

2.2 Calcium currents (based on Joux et al. 2001)

2.2.1 High threshold current with no or very slow inactivation (N-type)

$$ \begin{array}{*{20}l} {{I_{{{\text{Ca}},N,i}} = g_{{{\text{Ca}},N,i}} m_{{N,i}} {\left( {V_{i} - E_{{{\text{Ca}},i}} } \right)};} \hfill} \\ {{{{\text{d}}m_{{N,i}} } \mathord{\left/ {\vphantom {{{\text{d}}m_{{N,i}} } {{\text{d}}t}}} \right. \kern-\nulldelimiterspace} {{\text{d}}t} = {{\left( {m_{{N\infty ,i}} - m_{{N,i}} } \right)}} \mathord{\left/ {\vphantom {{{\left( {m_{{N\infty ,i}} - m_{{N,i}} } \right)}} {t_{{{\text{mN}}}} }}} \right. \kern-\nulldelimiterspace} {t_{{{\text{mN}}}} };} \hfill} \\ {{m_{{N\infty ,i}} = {1.0} \mathord{\left/ {\vphantom {{1.0} {{\left( {1.0 + \exp {\left( { - {{\left( {V_{i} + 11.0} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} + 11.0} \right)}} {4.2}}} \right. \kern-\nulldelimiterspace} {4.2}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1.0 + \exp {\left( { - {{\left( {V_{i} + 11.0} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} + 11.0} \right)}} {4.2}}} \right. \kern-\nulldelimiterspace} {4.2}} \right)}} \right)}}.} \hfill} \\ \end{array} $$

2.2.2 Lower threshold (L-type) current

$$ \begin{array}{*{20}l} {{I_{{{\text{Ca}},L,i}} = g_{{{\text{Ca}},L,i}} m_{{L,ih}} L_{{\infty ,i}} {\left( {V_{i} - E_{{{\text{Ca}},i}} } \right)};} \hfill} \\ {{{{\text{d}}m_{{L,i}} } \mathord{\left/ {\vphantom {{{\text{d}}m_{{L,i}} } {{\text{d}}t}}} \right. \kern-\nulldelimiterspace} {{\text{d}}t} = {{\left( {m_{{L\infty ,i}} - m_{{L,i}} } \right)}} \mathord{\left/ {\vphantom {{{\left( {m_{{L\infty ,i}} - m_{{L,i}} } \right)}} {\tau _{{mL}} }}} \right. \kern-\nulldelimiterspace} {\tau _{{mL}} };} \hfill} \\ {{m_{{L\infty ,{\text{s}}}} = {0.5} \mathord{\left/ {\vphantom {{0.5} {{\left( {1.0 + \exp {\left( {{ - {\left( {V_{{\text{s}}} + 27.0} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{{\text{s}}} + 27.0} \right)}} {4.5}}} \right. \kern-\nulldelimiterspace} {4.5}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1.0 + \exp {\left( {{ - {\left( {V_{{\text{s}}} + 27.0} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{{\text{s}}} + 27.0} \right)}} {4.5}}} \right. \kern-\nulldelimiterspace} {4.5}} \right)}} \right)}} + {0.5} \mathord{\left/ {\vphantom {{0.5} {{\left( {1.0 + \exp {\left( {{ - {\left( {V_{{\text{s}}} + 11.4} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{{\text{s}}} + 11.4} \right)}} {2.0}}} \right. \kern-\nulldelimiterspace} {2.0}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1.0 + \exp {\left( {{ - {\left( {V_{{\text{s}}} + 11.4} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{{\text{s}}} + 11.4} \right)}} {2.0}}} \right. \kern-\nulldelimiterspace} {2.0}} \right)}} \right)}};} \hfill} \\ {{m_{{L\infty ,{\text{pd}}}} = {0.4} \mathord{\left/ {\vphantom {{0.4} {{\left( {1.0 + \exp {\left( {{ - {\left( {V_{{{\text{pd}}}} + 27.0} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{{{\text{pd}}}} + 27.0} \right)}} {4.5}}} \right. \kern-\nulldelimiterspace} {4.5}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1.0 + \exp {\left( {{ - {\left( {V_{{{\text{pd}}}} + 27.0} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{{{\text{pd}}}} + 27.0} \right)}} {4.5}}} \right. \kern-\nulldelimiterspace} {4.5}} \right)}} \right)}} + {0.6} \mathord{\left/ {\vphantom {{0.6} {{\left( {1.0 + \exp {\left( { - {{\left( {V_{{{\text{pd}}}} {\text{ + }}11.4} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{{{\text{pd}}}} {\text{ + }}11.4} \right)}} {2.0}}} \right. \kern-\nulldelimiterspace} {2.0}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1.0 + \exp {\left( { - {{\left( {V_{{{\text{pd}}}} {\text{ + }}11.4} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{{{\text{pd}}}} {\text{ + }}11.4} \right)}} {2.0}}} \right. \kern-\nulldelimiterspace} {2.0}} \right)}} \right)}};} \hfill} \\ {{m_{{L\infty ,{\text{sd}}}} = {0.3} \mathord{\left/ {\vphantom {{0.3} {}}} \right. \kern-\nulldelimiterspace} {}{\left( {1.0 + \exp {\left( {{ - {\left( {V_{{{\text{sd}}}} + 27.0} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{{{\text{sd}}}} + 27.0} \right)}} {4.5}}} \right. \kern-\nulldelimiterspace} {4.5}} \right)}} \right)} + {0.7} \mathord{\left/ {\vphantom {{0.7} {{\left( {1.0 + \exp {\left( {{ - {\left( {V_{{{\text{sd}}}} + 11.4} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{{{\text{sd}}}} + 11.4} \right)}} {2.0}}} \right. \kern-\nulldelimiterspace} {2.0}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1.0 + \exp {\left( {{ - {\left( {V_{{{\text{sd}}}} + 11.4} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{{{\text{sd}}}} + 11.4} \right)}} {2.0}}} \right. \kern-\nulldelimiterspace} {2.0}} \right)}} \right)}};} \hfill} \\ {{h_{{L\infty ,i}} = {0.2\;K^{4}_{{M,L1}} } \mathord{\left/ {\vphantom {{0.2\;K^{4}_{{M,L1}} } {{\left( {K^{4}_{{M,L1}} + {\left[ {{\text{Ca}}^{{2 + }} } \right]}^{4}_{{{\text{in}},i}} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {K^{4}_{{M,L1}} + {\left[ {{\text{Ca}}^{{2 + }} } \right]}^{4}_{{{\text{in}},i}} } \right)}} + {0.8\;K_{{M,L2}} } \mathord{\left/ {\vphantom {{0.8\;K_{{M,L2}} } {{\left( {K_{{M,L2}} + {\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {K_{{M,L2}} + {\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} } \right)}}.} \hfill} \\ \end{array} $$

2.3 Potassium currents

2.3.1 Delayed rectifier current (based on Luther and Tasker 2000)

$$ \begin{array}{*{20}l} {{I_{{KDR,i}} = g_{{KDR,i}} n^{3} i{\left( {V_{i} - E_{K} } \right)};} \hfill} \\ {{{{\text{d}}n_{i} } \mathord{\left/ {\vphantom {{{\text{d}}n_{i} } {{\text{d}}t}}} \right. \kern-\nulldelimiterspace} {{\text{d}}t} = {{\left( {n_{{\infty ,i}} - n_{i} } \right)}} \mathord{\left/ {\vphantom {{{\left( {n_{{\infty ,i}} - n_{i} } \right)}} {\tau _{n} }}} \right. \kern-\nulldelimiterspace} {\tau _{n} };} \hfill} \\ {{n_{{\infty ,i}} = 1 \mathord{\left/ {\vphantom {1 {{\left( {1 + \exp {\left( {{ - {\left( {V_{i} + 18.3} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{i} + 18.3} \right)}} 9}} \right. \kern-\nulldelimiterspace} 9} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left( {{ - {\left( {V_{i} + 18.3} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{i} + 18.3} \right)}} 9}} \right. \kern-\nulldelimiterspace} 9} \right)}} \right)}};} \hfill} \\ {{\tau _{{n,i}} = {3.6} \mathord{\left/ {\vphantom {{3.6} {{\left( {1 + \exp {\left( {{{\left( {V_{i} - 3.0} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} - 3.0} \right)}} {5.0}}} \right. \kern-\nulldelimiterspace} {5.0}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left( {{{\left( {V_{i} - 3.0} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} - 3.0} \right)}} {5.0}}} \right. \kern-\nulldelimiterspace} {5.0}} \right)}} \right)}} + {1.6} \mathord{\left/ {\vphantom {{1.6} {{\left( {1 + \exp {\left( {{V_{i} } \mathord{\left/ {\vphantom {{V_{i} } {10.0}}} \right. \kern-\nulldelimiterspace} {10.0}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left( {{V_{i} } \mathord{\left/ {\vphantom {{V_{i} } {10.0}}} \right. \kern-\nulldelimiterspace} {10.0}} \right)}} \right)}} + {5.2} \mathord{\left/ {\vphantom {{5.2} {{\left( {1 + \exp {\left( {{ - {\left( {V_{i} + 65.0} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{i} + 65.0} \right)}} {6.0}}} \right. \kern-\nulldelimiterspace} {6.0}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left( {{ - {\left( {V_{i} + 65.0} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{i} + 65.0} \right)}} {6.0}}} \right. \kern-\nulldelimiterspace} {6.0}} \right)}} \right)}} - 4.0.} \hfill} \\ \end{array} $$

2.3.2 A-current (based on Luther and Tasker 2000)

$$ \begin{array}{*{20}l} {{I_{{A,i}} = g_{{A,i}} p^{4} _{i} q_{i} {\left( {V_{i} - E_{K} } \right)};} \hfill} \\ {{{{\text{d}}p_{i} } \mathord{\left/ {\vphantom {{{\text{d}}p_{i} } {{\text{d}}t}}} \right. \kern-\nulldelimiterspace} {{\text{d}}t} = {{\left( {p_{{\infty ,i}} - p_{i} } \right)}} \mathord{\left/ {\vphantom {{{\left( {p_{{\infty ,i}} - p_{i} } \right)}} {\tau _{{p,i}} }}} \right. \kern-\nulldelimiterspace} {\tau _{{p,i}} };} \hfill} \\ {{p_{{\infty ,i}} = 1 \mathord{\left/ {\vphantom {1 {{\left( {1 + \exp {\left( {{ - {\left( {V_{i} + 46.8} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{i} + 46.8} \right)}} {9.3}}} \right. \kern-\nulldelimiterspace} {9.3}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left( {{ - {\left( {V_{i} + 46.8} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{i} + 46.8} \right)}} {9.3}}} \right. \kern-\nulldelimiterspace} {9.3}} \right)}} \right)}};} \hfill} \\ {{{{\text{d}}q_{i} } \mathord{\left/ {\vphantom {{{\text{d}}q_{i} } {{\text{d}}t}}} \right. \kern-\nulldelimiterspace} {{\text{d}}t} = {{\left( {q_{{\infty ,i}} - q_{i} } \right)}} \mathord{\left/ {\vphantom {{{\left( {q_{{\infty ,i}} - q_{i} } \right)}} {\tau _{{q,i}} }}} \right. \kern-\nulldelimiterspace} {\tau _{{q,i}} };} \hfill} \\ {{q_{{\infty ,i}} = 1 \mathord{\left/ {\vphantom {1 {{\left( {1 + \exp {\left( {{{\left( {V_{i} + 80.1} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} + 80.1} \right)}} {8.8}}} \right. \kern-\nulldelimiterspace} {8.8}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left( {{{\left( {V_{i} + 80.1} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} + 80.1} \right)}} {8.8}}} \right. \kern-\nulldelimiterspace} {8.8}} \right)}} \right)}};} \hfill} \\ {{\tau _{{p,i}} = 0.2 + {6.4} \mathord{\left/ {\vphantom {{6.4} {{\left( {\exp {\left( {{{\left( {V_{i} + 68.0} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} + 68.0} \right)}} {27.0}}} \right. \kern-\nulldelimiterspace} {27.0}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {\exp {\left( {{{\left( {V_{i} + 68.0} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} + 68.0} \right)}} {27.0}}} \right. \kern-\nulldelimiterspace} {27.0}} \right)}} \right)}} + 4.0\exp {\left( {{ - {\left( {V_{i} + 130.0} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{i} + 130.0} \right)}} {35.0}}} \right. \kern-\nulldelimiterspace} {35.0}} \right)} + {0.4} \mathord{\left/ {\vphantom {{0.4} {{\left( {1 + \exp {\left( {{{\left( {V_{i} - 25.0} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} - 25.0} \right)}} {4.0}}} \right. \kern-\nulldelimiterspace} {4.0}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left( {{{\left( {V_{i} - 25.0} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} - 25.0} \right)}} {4.0}}} \right. \kern-\nulldelimiterspace} {4.0}} \right)}} \right)}};} \hfill} \\ {{\tau _{{q,i}} = {350} \mathord{\left/ {\vphantom {{350} {{\left( {1 + \exp {\left( {{{\left( {V_{i} + 120} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} + 120} \right)}} {10.0}}} \right. \kern-\nulldelimiterspace} {10.0}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left( {{{\left( {V_{i} + 120} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} + 120} \right)}} {10.0}}} \right. \kern-\nulldelimiterspace} {10.0}} \right)}} \right)}} + {7.5} \mathord{\left/ {\vphantom {{7.5} {{\left( {1 + \exp {\left( {{{\left( {V_{i} + 8.0} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} + 8.0} \right)}} {5.0}}} \right. \kern-\nulldelimiterspace} {5.0}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left( {{{\left( {V_{i} + 8.0} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} + 8.0} \right)}} {5.0}}} \right. \kern-\nulldelimiterspace} {5.0}} \right)}} \right)}} + {12.5} \mathord{\left/ {\vphantom {{12.5} {{\left( {1 + \exp {\left( {{{\left( {V_{i} - 2.0} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} - 2.0} \right)}} {5.0}}} \right. \kern-\nulldelimiterspace} {5.0}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left( {{{\left( {V_{i} - 2.0} \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{i} - 2.0} \right)}} {5.0}}} \right. \kern-\nulldelimiterspace} {5.0}} \right)}} \right)}} + 1.0.} \hfill} \\ \end{array} $$

2.3.3 SOR current (for the model OT neuron only; based on Stern and Armstrong 1995)

$$ \begin{array}{*{20}l} {{I_{{{\text{SOR}},i}} = g_{{{\text{SOR}},i}} m_{{{\text{SOR}},i}} {\left( {V_{i} - E_{K} } \right)};} \hfill} \\ {{{{\text{d}}m_{{{\text{SOR}},i}} } \mathord{\left/ {\vphantom {{{\text{d}}m_{{{\text{SOR}},i}} } {{\text{d}}t}}} \right. \kern-\nulldelimiterspace} {{\text{d}}t} = {{\left( {m_{{{\text{SOR}}\infty ,i}} - m_{{{\text{SOR}},i}} } \right)}} \mathord{\left/ {\vphantom {{{\left( {m_{{{\text{SOR}}\infty ,i}} - m_{{{\text{SOR}},i}} } \right)}} {\tau _{{m,{\text{SOR}}}} }}} \right. \kern-\nulldelimiterspace} {\tau _{{m,{\text{SOR}}}} };} \hfill} \\ {{m_{{{\text{SOR}}\infty ,i}} = {1.0} \mathord{\left/ {\vphantom {{1.0} {{\left( {1.0 + \exp {\left( {{ - {\left( {V_{i} + 60.0} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{i} + 60.0} \right)}} {4.0}}} \right. \kern-\nulldelimiterspace} {4.0}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1.0 + \exp {\left( {{ - {\left( {V_{i} + 60.0} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{i} + 60.0} \right)}} {4.0}}} \right. \kern-\nulldelimiterspace} {4.0}} \right)}} \right)}}.} \hfill} \\ \end{array} $$

2.3.4 Ca²⁺-sensitive K⁺ currents (In the equations, [Ca²⁺] is in mM)

2.3.4.1 SK current (based on Vergara et al. 1998; Xia et al. 1998)

$$ \begin{array}{*{20}l} {{I_{{{\text{SK}},i}} = g_{{,{\text{SK}},i}} m_{{{\text{SK}}\infty ,i}} {\left( {V_{i} - E_{K} } \right)};} \hfill} \\ {{m_{{{\text{SK}}\infty ,i}} = 1 \mathord{\left/ {\vphantom {1 {{\left( {1 + {\left( {{K_{{m,{\text{SK}}}} } \mathord{\left/ {\vphantom {{K_{{m,{\text{SK}}}} } {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} }}} \right. \kern-\nulldelimiterspace} {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} }} \right)}^{{4.5}} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + {\left( {{K_{{m,{\text{SK}}}} } \mathord{\left/ {\vphantom {{K_{{m,{\text{SK}}}} } {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} }}} \right. \kern-\nulldelimiterspace} {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} }} \right)}^{{4.5}} } \right)}}.} \hfill} \\ \end{array} $$

2.3.4.2 BK current (based on Dopico et al. 1999)

$$ \begin{array}{*{20}l} {{I_{{BK,i}} = g_{{{\text{BK}},i}} m_{{{\text{BK}}\infty ,i}} {\left( {V_{i} - E_{K} } \right)};} \hfill} \\ {{m_{{{\text{BK}}\infty ,i}} = 1 \mathord{\left/ {\vphantom {1 {{\left( {1 + \exp {\left( {{ - {\left( {V_{i} - {\left( { - 138.37\;\log _{{10}} {\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{BK}},i}} } \right)} - 463} \right)}} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{i} - {\left( { - 138.37\;\log _{{10}} {\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{BK}},i}} } \right)} - 463} \right)}} \right)}} {11}}} \right. \kern-\nulldelimiterspace} {11}} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left( {{ - {\left( {V_{i} - {\left( { - 138.37\;\log _{{10}} {\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{BK}},i}} } \right)} - 463} \right)}} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{i} - {\left( { - 138.37\;\log _{{10}} {\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{BK}},i}} } \right)} - 463} \right)}} \right)}} {11}}} \right. \kern-\nulldelimiterspace} {11}} \right)}} \right)}}.} \hfill} \\ \end{array} $$

2.4 CAN current

$$ \begin{array}{*{20}l} {{I_{{{\text{CAN}},i}} = g_{{{\text{CAN}},i}} m_{{{\text{CAN}},i}} {\left( {2V_{i} - E_{{{\text{Na}}}} - E_{K} } \right)};} \hfill} \\ {{m_{{{\text{CAN}},i}} = A_{i} {\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} } \right)} \cdot B_{i} {\left( {V_{i} ,{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} } \right)};} \hfill} \\ {{A_{i} {\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} } \right)} = {{\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} } \right)}^{2} } \mathord{\left/ {\vphantom {{{\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} } \right)}^{2} } {{\left( {{\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} } \right)}^{2} + K^{2}_{{d,{\text{CAN}}}} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {{\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} } \right)}^{2} + K^{2}_{{d,{\text{CAN}}}} } \right)}};} \hfill} \\ {{B_{i} {\left( {V_{i} ,{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} } \right)} = 1 \mathord{\left/ {\vphantom {1 {{\left( {1 + \exp {\left( {{ - {\left( {V_{i} - V_{{{\text{CAN}},h}} \cdot S_{i} {\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} } \right)}} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{i} - V_{{{\text{CAN}},h}} \cdot S_{i} {\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} } \right)}} \right)}} 3}} \right. \kern-\nulldelimiterspace} 3} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left( {{ - {\left( {V_{i} - V_{{{\text{CAN}},h}} \cdot S_{i} {\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} } \right)}} \right)}} \mathord{\left/ {\vphantom {{ - {\left( {V_{i} - V_{{{\text{CAN}},h}} \cdot S_{i} {\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} } \right)}} \right)}} 3}} \right. \kern-\nulldelimiterspace} 3} \right)}} \right)}};} \hfill} \\ {{S_{i} = {1.0} \mathord{\left/ {\vphantom {{1.0} {{\left( {1 + \exp {\left( {{{\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} - a_{1} } \right)}} \mathord{\left/ {\vphantom {{{\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} - a_{1} } \right)}} {b_{1} }}} \right. \kern-\nulldelimiterspace} {b_{1} }} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left( {{{\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} - a_{1} } \right)}} \mathord{\left/ {\vphantom {{{\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} - a_{1} } \right)}} {b_{1} }}} \right. \kern-\nulldelimiterspace} {b_{1} }} \right)}} \right)}} + {1.2} \mathord{\left/ {\vphantom {{1.2} {{\left( {1 + \exp {\left( {{{\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} - a_{2} } \right)}} \mathord{\left/ {\vphantom {{{\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} - a_{2} } \right)}} {b_{2} }}} \right. \kern-\nulldelimiterspace} {b_{2} }} \right)}} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {1 + \exp {\left( {{{\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} - a_{2} } \right)}} \mathord{\left/ {\vphantom {{{\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{in}},i}} - a_{2} } \right)}} {b_{2} }}} \right. \kern-\nulldelimiterspace} {b_{2} }} \right)}} \right)}} - 5.9.} \hfill} \\ \end{array} $$

2.5 Linear leakage current

$$ \begin{array}{*{20}l} {{I_{{L,i}} = I_{{L,{\text{Na}},i}} + I_{{L,K,i}} ;} \hfill} & {{I_{{L,K,i}} = g_{{L,K,i}} {\left( {V_{i} - E_{K} } \right)};} \hfill} & {{I_{{L,{\text{Na}},i}} = g_{{L,{\text{Na}},i}} {\left( {V_{i} - E_{{{\text{Na}}}} } \right)}} \hfill} \\ \end{array} $$

Calcium dynamics

3.1 Calcium dynamics in the cytoplasm

$$ {{\text{d}}{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{i} } \mathord{\left/ {\vphantom {{{\text{d}}{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{i} } {{\text{d}}t}}} \right. \kern-\nulldelimiterspace} {{\text{d}}t} = 2f_{{{\text{Ca}}}} {\left( {{ - {\left( {\Sigma I_{{{\text{Ca}},j,i}} } \right)}} \mathord{\left/ {\vphantom {{ - {\left( {\Sigma I_{{{\text{Ca}},j,i}} } \right)}} {{\left( {{\text{d}}_{i} F} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {{\text{d}}_{i} F} \right)}} - U_{i} {\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{i} - {\left[ {{\text{Ca}}^{{2 + }} } \right]}_{r} } \right)}} \right)} $$

Subscript “j” represents the L- or N-type Ca²⁺ current in the compartment.

3.2 Calcium dynamics in the BK channel subdomains

$$ {{\text{d}}{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{BK}},i}} } \mathord{\left/ {\vphantom {{{\text{d}}{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{BK}},i}} } {{\text{d}}t}}} \right. \kern-\nulldelimiterspace} {{\text{d}}t} = 2f_{{{\text{Ca}},{\text{BK}}}} {\left( {{ - I_{{{\text{Ca}},N,j}} } \mathord{\left/ {\vphantom {{ - I_{{{\text{Ca}},N,j}} } {{\left( {r{\text{d}}_{i} F} \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {r{\text{d}}_{i} F} \right)}} - K_{{{\text{BK}},i}} {\left( {{\left[ {{\text{Ca}}^{{2 + }} } \right]}_{{{\text{BK}},i}} - {\left[ {{\text{Ca}}^{{2 + }} } \right]}_{r} } \right)}} \right)}, $$

Here, “i” can be either “s” (soma) or “pd” (primary dendrite).

Intercompartmental coupling currents

$$ \begin{array}{*{20}c} {I_{{{\text{sd - pd}}}} = g_{{{\text{sd - pd}}}} {\left( {V_{{{\text{sd}}}} - V_{{{\text{pd}}}} } \right)};\;I_{{{\text{pd - sd}}}} = g_{{{\text{pd}}}} {\left( {V_{{{\text{pd}}}} - V_{{{\text{sd}}}} } \right)};\;I_{{{\text{pd - s}}}} = g_{{{\text{pd - s}}}} {\left( {V_{{{\text{pd}}}} - V_{{\text{s}}} } \right)};} \\ {I_{{{\text{s - pd}}}} = g_{{{\text{s - pd}}}} {\left( {V_{{\text{s}}} - V_{{{\text{pd}}}} } \right)};} \\ {G_{{{\text{s - pd}}}} = {10^{2} \pi d^{2}_{{{\text{pd}}}} d^{2}_{{\text{s}}} } \mathord{\left/ {\vphantom {{10^{2} \pi d^{2}_{{{\text{pd}}}} d^{2}_{{\text{s}}} } {2\;Ra{\left( {L_{{{\text{pd}}}} {\text{d}}^{2}_{{\text{s}}} + L_{{\text{s}}} {\text{d}}^{2}_{{{\text{pd}}}} } \right)}}}} \right. \kern-\nulldelimiterspace} {2\;Ra{\left( {L_{{{\text{pd}}}} {\text{d}}^{2}_{{\text{s}}} + L_{{\text{s}}} {\text{d}}^{2}_{{{\text{pd}}}} } \right)}};} \\ {G_{{{\text{pd - sd}}}} = {10^{2} \pi {\text{d}}^{2}_{{{\text{pd}}}} {\text{d}}^{2}_{{{\text{sd}}}} } \mathord{\left/ {\vphantom {{10^{2} \pi {\text{d}}^{2}_{{{\text{pd}}}} {\text{d}}^{2}_{{{\text{sd}}}} } {2\;Ra{\left( {L_{{{\text{pd}}}} {\text{d}}^{2}_{{{\text{sd}}}} + L_{{{\text{sd}}}} {\text{d}}^{2}_{{{\text{pd}}}} } \right)}}}} \right. \kern-\nulldelimiterspace} {2\;Ra{\left( {L_{{{\text{pd}}}} {\text{d}}^{2}_{{{\text{sd}}}} + L_{{{\text{sd}}}} {\text{d}}^{2}_{{{\text{pd}}}} } \right)}};} \\ {g_{{{\text{s - pd}}}} = 2 \cdot {10^{8} \;G_{{{\text{s - pd}}}} } \mathord{\left/ {\vphantom {{10^{8} \;G_{{{\text{s - pd}}}} } {{\left( {\pi {\text{d}}_{{\text{s}}} L_{{\text{s}}} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {\pi {\text{d}}_{{\text{s}}} L_{{\text{s}}} } \right)}};\;g_{{{\text{pd - s}}}} = {10^{8} \;G_{{{\text{s - pd}}}} } \mathord{\left/ {\vphantom {{10^{8} \;G_{{{\text{s - pd}}}} } {{\left( {\pi {\text{d}}_{{{\text{pd}}}} L_{{{\text{pd}}}} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {\pi {\text{d}}_{{{\text{pd}}}} L_{{{\text{pd}}}} } \right)}};} \\ {g_{{{\text{pd - sd}}}} = 4 \cdot {10^{8} \;G_{{{\text{pd - sd}}}} } \mathord{\left/ {\vphantom {{10^{8} \;G_{{{\text{pd - sd}}}} } {{\left( {\pi {\text{d}}_{{{\text{pd}}}} L_{{{\text{pd}}}} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {\pi {\text{d}}_{{{\text{pd}}}} L_{{{\text{pd}}}} } \right)}};\;g_{{{\text{sd - pd}}}} = {10^{8} \;G_{{{\text{pd - sd}}}} } \mathord{\left/ {\vphantom {{10^{8} \;G_{{{\text{pd - sd}}}} } {{\left( {\pi {\text{d}}_{{{\text{sd}}}} L_{{{\text{sd}}}} } \right)}}}} \right. \kern-\nulldelimiterspace} {{\left( {\pi {\text{d}}_{{{\text{sd}}}} L_{{{\text{sd}}}} } \right)}}.} \\ \end{array} $$

Applied current: I _stim = I _ST/(πd_s L _s).