On the mechanisms underlying the depolarization block in the spiking dynamics of CA1 pyramidal neurons

Abstract

Under sustained input current of increasing strength neurons eventually stop firing, entering a depolarization block. This is a robust effect that is not usually explored in experiments or explicitly implemented or tested in models. However, the range of current strength needed for a depolarization block could be easily reached with a random background activity of only a few hundred excitatory synapses. Depolarization block may thus be an important property of neurons that should be better characterized in experiments and explicitly taken into account in models at all implementation scales. Here we analyze the spiking dynamics of CA1 pyramidal neuron models using the same set of ionic currents on both an accurate morphological reconstruction and on its reduction to a single-compartment. The results show the specific ion channel properties and kinetics that are needed to reproduce the experimental findings, and how their interplay can drastically modulate the neuronal dynamics and the input current range leading to a depolarization block. We suggest that this can be one of the rate-limiting mechanisms protecting a CA1 neuron from excessive spiking activity.

Appendix

1.1 Ionic currents

The full morphological model includes the following 17 types of ionic channels, most of them distributed non-uniformly along the somatodendritic region. More precisely, the K _M and K _A potassium currents are those described by Shah et al. (2008), whereas the other types of currents have been distributed as in model of Poirazi et al. (2003b).

• Soma: a leak current, a transient sodium (NaT) current, a delay-rectifier potassium (K _DR) current, an A-type potassium current (K _A), a M type potassium current (K _M), a mixed conductance hyper-polarization activated h-current, three types of voltage dependent calcium currents (namely LVA T-type current, a HVA R-type current, a HVA L-type current), two types of calcium dependent potassium currents (a slow AHP current and a medium fast AHP current);

• Axon: a NaT current, a K _DR current, a leak current, a K _A current and K _M current;

• Basal dendrites: a NaT current, a K _DR current, a leak current, a K _A current and a h-current;

• Apical trunk: all currents included in the soma with the exception of K _M current and the insertion of a K _A current (the HVA R and L type are different from those in soma);

• Apical dendrites: all currents included in the apical trunk with an insertion of a persistent sodium current.

Ionic currents were modeled following a Hodgkin-Huxley-like formalism as follows:

$$ I_{j}=\bar{g}_{j}m_{j}^{a_{j}}h_{j}^{b_{j}}(V-E_{j}), $$

(2)

where $\bar{g}_{j}$ represents the maximal ionic conductance, (m _j, h _j) and (a _j, b _j) are the gating variables for activation and inactivation and their exponents, respectively, and E _j is the reversal potential associated with the particular ion or ions that make up the current. The dynamics of the gating variables (m _j , h _j ) is governed by an ODE of the form

$$ \frac{d\chi}{dt}= \frac{\chi_{\infty}-\chi}{\tau_{\chi}} $$

(3)

where the activation and inactivation steady-state functions χ _∞, and their time constant τ _χ are given by

$$ \chi _{\infty }=\frac{\alpha _{\chi }(V)}{\alpha _{\chi }(V)+\beta _{\chi }(V)},\quad \chi _{\infty }=\frac{1}{1+\exp \left[ \left( V-V_{\chi }^{1/2}\right) /k_{\chi }\right] }, $$

(4)

$$ \tau_{\chi } =\frac{Q}{\alpha _{\chi }(V)+\beta _{\chi }(V)}, $$

(5)

$$ \tau _{\chi } =\tau _{\chi }^{0}+\frac{\bar{\tau}_{\chi }}{G_{\chi }\left(\exp \left[ \gamma _{\chi }\left( V-V_{\chi }^{1/2}\right) /\widetilde{k} _{\chi }\right] +\exp \left[ \left( \gamma _{\chi }-1\right) \left( V-V_{\chi }^{1/2}\right) /\widetilde{k}_{\chi }\right] \right)}, $$

(6)

In the following subsections we report only the kinetic details of the currents directly involved with the results discussed in this paper. The parameters of all the other currents were not modified from their original values.

1.1.1 Transient NaT current

The NaT current was implemented according to Shah et al. (2008), so Eq. (2) becomes:

$$I_{\rm NaT}= \overline{g}_{\rm NaT} \; m_{\rm NaT}^3 \;h_{\rm NaT}(V-50). $$

(7)

The dynamics of the gating variable m _NaT is described by Eqs. (3), (4)₁ and (5) as follows

$$ \begin{array}{rll} \alpha _{\rm m_{\rm NaT}} &=&\left\{ \begin{array}{ll} \dfrac{0.4\;(V-V_{\rm m_{\rm NaT}}^{1/2})}{1-\exp[-(V-V_{\rm m_{\rm NaT}}^{1/2})/k_{\rm m_{\rm NaT}}]},\\\\ |V-V_{\rm m_{\rm NaT}}^{1/2}|>10^{-6}\;\rm mV \\\\ 0.4 \;k_{\rm m_{\rm NaT}}, & \mathrm{otherwise} \end{array} \right. , \\ \beta _{\rm m_{\rm NaT}} &=&\left\{ \begin{array}{ll} \dfrac{-0.124\;(V-V_{\rm m_{\rm NaT}}^{1/2})}{1-\exp[(V-V_{\rm m_{\rm NaT}}^{1/2})/k_{\rm m_{\rm NaT}}]},\\\\ |V_{\rm m_{\rm NaT}}^{1/2}-V|>10^{-6}\;mV \\\\ 0.124\; k_{\rm m_{\rm NaT}}, & \mathrm{otherwise} \end{array} \right. , \\ \tau_{\rm m_{\rm NaT}} &=&\left\{ \begin{array}{ll} \dfrac{3}{\alpha _{\rm m_{\rm NaT} }(V)+\beta _{\rm m_{\rm NaT} }(V)},\\\\ \dfrac{3}{\alpha _{\rm m_{\rm NaT} }(V)+\beta _{\rm m_{\rm NaT} }(V)}> 0.02 \rm \;ms\\\\ 0.02 \rm \;ms, & \mathrm{otherwise} \end{array} \right. , \end{array} $$

The dynamics of the gating variable h _NaT is described by Eqs. (3), (4)₂, and (5), i.e.,

$$ \begin{array}{rll} (h_{\rm NaT})_{\infty}&=&\frac{1}{1+\exp[(V-V_{\rm h_{\rm NaT}}^{1/2})/k_{\rm h_{\rm NaT}}]}, \\[6pt] \alpha _{\rm h_{\rm NaT}} &=&\left\{ \begin{array}{ll} \dfrac{ 0.03 \;(V+45)}{1- \exp[-(V+45)/1.5]},\\\\ |V-V_{\rm h_{\rm NaT}}^{1/2}|>10^{-6}\rm \;mV \\\\ 0.045 , & \mathrm{otherwise} \end{array} \right. , \\ \beta _{\rm h_{\rm NaT}} &=&\left\{ \begin{array}{ll} \dfrac{- 0.01 \; (V+45)}{1- \exp[(V+45)/1.5]},\\\\ |V_{\rm h_{\rm NaT}}^{1/2}-V|>10^{-6} \rm \;mV \\\\ 0.015 , & \mathrm{otherwise} \end{array} \right. \end{array} $$

All the parameters written in this section are listed in Table 3.

1.1.2 The delayed rectifier K ⁺ current

The delayed rectifier K ⁺ current ($I_{\rm K_{\rm DR}}$) taken from Shah et al. (2008) is given by:

$$ I_{K_{DR}}= \overline{g}_{K_{DR}} \; m_{K_{DR}}\; (V+77) $$

(8)

The dynamics of $m_{\rm K_{\rm DR}}$ is described by Eqs. (3), (4)₂ and (6), with $V_{\rm m_{\rm K_{\rm DR}}}^{1/2}=13\;\rm mV$, $k_{\rm K_{\rm DR}}=-8.824\rm \;mV$, $\tau_{\rm m_{\rm K_{\rm DR}}}^{0}=0\rm \;ms$, $\gamma_{\rm m_{\rm K_{\rm DR}}}=0.7$, $G_{\rm K_{\rm DR}}=1$, $\bar{\tau}_{\rm m_{\rm K_{\rm DR}}}=\frac{1}{0.02}\rm \; ms$, $\widetilde{k}_{\rm s_{\rm NaT}}=8.824\rm \;mV$, and a minimum value for $\tau_{\rm m_{\rm K_{\rm DR}}}$ of 2 ms.

1.1.3 The potassium M-type current

The M current ($I_{K_{M}}$) was modeled according to Shah et al. (2008), namely

$$ I_{K_{M}}= \overline{g}_{K_{M}} \; m_{K_{M}}\; (V+77) $$

(9)

The dynamics of the gating variable $m_{K_{M}}$ is described by Eqs. (3), (6) and (4)₂ with the following formula:

$$ (m_{K_{M}})_{\infty}=\frac{1}{1+\exp[(V-V_{m_{K_{M}}}^{1/2})/(k_{m_{K_{M}}}]} $$

(10)

Some parameters are listed in Table 3 and the others are set as follows: $\tau_{m_{K_{M}}}^{0}=60\rm \; ms$, $G_{m_{K_{M}}}=1$.

1.1.4 The m-type Ca-dependent potassium current, mAHP

The medium AHP current I _mAHP, from Moczydlowski and Latorre (1983), is given by

$$ I_{\rm mAHP}= \overline{g}_{\rm mAHP} \; m_{\rm mAHP}\; (V+77) $$

(11)

The dynamics of the gating variable m _mAHP is described by Eqs. (3), (4)₁ and (5). The α _m and β _m are given by:

$$ \begin{array}{rll}\alpha_{\rm mAHP}&=&\frac{0.48} {1+ \frac{0.18}{[Ca^{2+}]_{i}}\exp[-2 d_{1} V \cdot 37.775]}, \\ \beta_{\rm mAHP}&=&\frac{0.28}{1+\frac{[Ca^{2+}]_{i}}{ 0.011\;\exp[-2 d_{2} V \cdot 37.775]}} \end{array} $$

(12)

$[Ca^{2+}]_{i}$ is the internal calcium concentration, and the other parameters are listed in Table 3.

1.2 Mathematical description of the somatic model

We recall that the single-compartment model takes into account only the ten somatic ionic currents listed in Section “Ionic currents”, i.e.,

• one transient Na ⁺ current I _NaT;

• three K ⁺ currents: one delayed rectifier $I_{\rm K_{\rm DR}}$, one muscarinic-sensitive $I_{K_{M}}$, and one A-type $I_{K_{A}}$;

• three Ca ^2 + currents: one LVA T-type current I _CaT, one HVA R-type current I _CaR, and one HVA L-type current I _CaL;

• one h current I _h ;

• two Ca ^2 + −activated K ⁺ currents: one slow I _sAHP, and one medium I _mAHP.

Then, the current balance equation for the somatic membrane potential V becomes

$$ \begin{aligned}[b] C_{m}\displaystyle \frac{dV}{dt}={}&-I_{\rm NaT}-I_{K_{\rm DR}}-I_{K_{M}}-I_{K_{A}}\\ &-I_{\rm CaT}-I_{\rm CaR}-I_{\rm CaL}-I_{h}\\ &-I_{\rm sAHP}-I_{\rm mAHP}-I_{\rm leak}+I_{\rm inj}. \end{aligned} $$

(13)

Thus, closing the membrane equation (13) with the ionic currents, the dynamics of gating variables, and the intracellular Ca ^2 + concentration, $\left[ Ca^{2+}\right]_{i}$, the model consists of a system of seventeen nonlinear ODEs in the unknown functions

$$ \begin{array} [t]{c} V,m_{\rm Na_{\rm T}},h_{\rm Na_{\rm T}},s_{\rm Na_{\rm T}},m_{\rm K_{\rm DR}},m_{\rm K_{\rm M}},m_{\rm K_{\rm A}},h_{\rm K_{\rm A}},m_{\rm CaT},h_{\rm CaT}, \\ m_{\rm CaR},h_{\rm CaR},m_{\rm CaL},m_{h},m_{\rm sAHP},m_{\rm mAHP},\left[ Ca^{2+}\right]_{i}. \end{array} $$

(14)

The model is analytically intractable, and solutions may be obtained only by numerical integration. The Cauchy problem for our model is:

$$ \left\{ \begin{array}{l} \displaystyle\frac{d\mathbf{X}}{dt}=\mathbf{f}\left( \mathbf{X},I_{\rm ext}\right) , \\\\ \mathbf{X}\left( 0,I_{\rm ext}\right) =\mathbf{X}_{0}\left( I_{\rm ext}\right), \end{array} \right. $$

(15)

where the components of the vector X are listed in Eq. (14), and $\mathbf{f}\left( \mathbf{X},I_{\rm ext}\right)$ is the vector whose components are the functions in the right-hand side of the ODEs. In addition, we set

$$ f_{1}\left(\mathbf{X},I_{\rm ext}\right)= I_{\rm ext}-G\left(\mathbf{X}\right). $$

(16)

In order to compare our dynamical model with experiments, we have to fix the values of all parameters and the external current. We set the potential V to the resting value V _rest and all the gating variables and intracellular Ca ^2 + concentration to the corresponding steady-state values, i.e.,

$$ X_{j,0}=\left\{ \begin{array}{ll} V_{\rm rest} & \quad j=1, \\\\ X_{j}^{\infty }\left( V_{\rm rest}\right) & \quad j=2,\ldots ,14, \\\\ m_{\rm sAHP}^{\infty }\left( \left[ Ca^{2+}\right]_{i} ^{\infty}\right) & \quad j=15, \\\\ m_{\rm mAHP}^{\infty }\left(V_{\rm rest}, \left[ Ca^{2+}\right]_{i} ^{\infty}\right) & \quad j=16, \\\\ \left[ Ca^{2+}\right]_{i} ^{\infty } & \quad j=17, \end{array} \right. $$

(17)

where $\left[ Ca^{2+}\right]_{i} ^{\infty }$ is a solution of the nonlinear equation

$$ f_{17}\left( V_{\rm rest},m_{\rm CaL}^{\infty },m_{\rm CaR}^{\infty },h_{\rm CaR}^{\infty },\left[ Ca^{2+} \right]_{i} \right) =0. $$

(18)

For the all simulations in Section 3.1.3 we have set the initial data as in Eq. (17) in which V _rest = − 70 mV.

A fundamental step in the qualitative analysis of our dynamical system is finding the equilibria. To start with, we calculate and characterize the stationary states X ^* which are defined by the condition $\mathbf{f}\left(\mathbf{X^*},I_{\rm ext}\right)=0$. As in the Hodgkin–Huxley-type models, a suitable numerical procedure was applied to reduce the vectorial equation $\mathbf{f}\left(\mathbf{X^*},I_{\rm ext}\right)=0$ to the nonlinear scalar equation

$$ f_{1}\left(\mathbf{X^*},I_{\rm ext}\right)= I_{\rm ext}-g\left(V^{*}\right)=0, $$

(19)

where $g\left(V\right)=G\left(V,X_{2}^{\infty }\left( V\right),...,X_{17}^{\infty }\left( V\right)\right)$. Owing to the presence of the calcium currents, the last three equations for the equilibria become

$$ \begin{aligned}[b] f_{15}\left(m_{\rm sAHP}^{*}, \left[ Ca^{2+}\right]_{i} ^{*}\right)&=0, \\ f_{16}\left(m_{\rm mAHP}^{*},V^{*}, \left[ Ca^{2+}\right]_{i} ^{*}\right)&=0, \\ f_{17}\left( V^{*},m_{\rm CaL}^{* },m_{\rm CaR}^{*},h_{\rm CaR}^{*},\left[ Ca^{2+} \right]_{i}^{*} \right) &=0, \end{aligned} $$

(20)

consequently a nonstandard numerical procedure was applied to derive Eq. (19). This last equation, which supplies the stationary point X ^* as a function of I _ext, can be solved only numerically. Linearizing the $\mathbf{f}\left(\mathbf{X^*},I_{\rm ext}\right)$ around the stationary states, we investigated their stability properties by varying the external current I _ext. The stability of the equilibria is characterized by the eigenvalues λ ₁, ..., λ ₁₇ of the usual Jacobian matrix

$$ J\left( I_{\rm ext}\right) =\frac{\partial f_{j}}{\partial X_{k}}\left( \mathbf{ X}^{\ast},I_{\rm ext}\right), $$

(21)

and all bifurcations diagrams are obtained using the procedure originally introduced by Troy for the Hodgkin–Huxley model in Troy (1974) (see also Hassard 1978; Troy 1978).