Spatial Memory Sequence Encoding and Replay During Modeled Theta and Ripple Oscillations

Abstract

Spatial learning involves the storage and replay of temporally ordered spatial information. The hippocampus is a key brain structure involved in spatial learning in rats. Temporally ordered spatial memories are encoded and replayed by the firing rate and phase of hippocampal pyramidal cells and inhibitory interneurons with respect to ongoing network theta and ripple oscillations paced by intra- and extrahippocampal areas. Theta oscillations (4–7 Hz) may contribute to memory formation, whereas fast ripple oscillations to temporally compressed forward and reverse replay of previously stored memories. Different classes of CA1 excitatory and inhibitory neurons and medial septal inhibitory neurons have been shown to differentially phase their activities with respect to theta and ripples. Understanding how the different hippocampal and extrahippocampal areas and their neuronal classes interact during these network oscillations and how they facilitate the storage and replay of spatiotemporal memories is of great importance. A computational model of the hippocampal CA1 microcircuit that uses biophysical representations of the major cell types, including pyramidal cells and four types of inhibitory interneurons, is extended. Inputs to the network come from the entorhinal cortex (EC), the CA3 Schaffer collaterals and the medial septum. A biophysical mechanism of spike timing-dependent plasticity (STDP) is used for learning spatial memory patterns in the correct order. The model addresses two important issues: (1) How are the storage and replay (forward and reverse) of temporally ordered memory patterns controlled in the CA1 microcircuit during theta and ripples? (2) What roles do the various types of inhibitory interneurons play in these processes?

Appendix: Mathematical Formalism

This Appendix contains the mathematical formalisms of the model cell types. Simulations were performed using the XPPAUT [23]. Data analysis was performed by MATLAB. Parameter units are measured in mV for potentials, μA/cm² for applied currents, mS/cm² for maximal conductances, and μF/cm² for capacitances.

CA1 Pyramidal Cell

The axonic (ax), somatic (s), proximal dendritic (pd) and distal dendritic (dd) compartments of the pyramidal neuron obey the following current balance equations

$$ C_{\text{m}} \frac{{{\text{d}}V_{\text{ax}} }}{{{\text{d}}t}} = I_{\text{L}} + I_{\text{Na,ax}} + I_{\text{K,ax}} + I_{\text{coup}} + I_{\text{syn}} + I_{\text{in}} $$

(4)

$$ C_{\text{m}} \frac{{{\text{d}}V_{\text{s}} }}{{{\text{d}}t}} = I_{\text{L}} + I_{\text{Na,s}} + I_{{{\text{K}}_{\text{dr,s}} }} + I_{\text{A,s}} + I_{\text{m,AHAP,s}} + I_{\text{CaL,s}} + I_{\text{coup}} + I_{\text{syn}} + I_{\text{in}} $$

(5)

$$ C_{\text{m}} \frac{{{\text{d}}V_{\text{pd}} }}{{{\text{d}}t}} = I_{\text{L}} + I_{\text{Na,d}} + I_{{{\text{K}}_{\text{dr,d}} }} + I_{\text{A,d}} + I_{\text{CaL,d}} + I_{\text{coup}} + I_{\text{syn}} + I_{\text{in}} $$

(6)

$$ C_{m} \frac{{{\text{d}}V_{\text{dd}} }}{{{\text{d}}t}} = I_{\text{L}} + I_{\text{Na,d}} + I_{{{\text{K}}_{\text{dr,d}} }} + I_{\text{A,d}} + I_{\text{CaL,d}} + I_{\text{coup}} + I_{\text{syn}} + I_{\text{in}} $$

(7)

where I _L is the leak current, I _Na is the sodium current, I _K is the delayed rectifier potassium current, I _A is the type-A potassium current [61], I _m,AHP is the medium Ca²⁺-activated K⁺ after-hyperpolarization current [61], I _CaL is the L-type Ca²⁺ current [61], I _coup is the electrical coupling between compartments, I _in is the injected current and I _syn is the synaptic current. Table 1 displays the ionic parameter values of the CA1 pyramidal cell.

Table 1 Pyramidal cell parameter values

The coupling currents for all compartments are

$$ I_{\text{coup}}^{\text{axon}} = g_{\text{axon,soma}} \cdot (V_{\text{soma}} - V_{\text{axon}} ) $$

(8)

$$ I_{\text{coup}}^{\text{soma}} = g_{\text{soma,axon}} \cdot (V_{\text{axon}} - V_{\text{soma}} ) + g_{{{\text{soma,dend}}_{\text{prox}} }} \cdot (V_{\text{pd}} - V_{\text{soma}} ) $$

(9)

$$ I_{\text{coup}}^{{{\text{dend}}_{\text{prox}} }} = g_{{{\text{soma,dend}}_{\text{prox}} }} \cdot (V_{\text{soma}} - V_{\text{pd}} ) + g_{{{\text{dend}}_{\text{dist}} , {\text{dend}}_{\text{prox}} }} (V_{\text{dd}} - V_{\text{pd}} ) $$

(10)

$$ I_{\text{coup}}^{{{\text{dend}}_{\text{dist}} }} = g_{{{\text{dend}}_{\text{prox}} , {\text{dend}}_{\text{dist}} }} (V_{\text{pd}} - V_{\text{dd}} ). $$

(11)

The leak current is described by

$$ I_{\text{L}} = g_{\text{L}} \cdot (V - V_{\text{L}} ) $$

(12)

where g _L is the leak conductance and V _L is the leak reversal potential.

The sodium current at the axon and soma is described by

$$ I_{\text{Na}} = - g_{\text{Na}} \cdot M_{\text{Na}}^{2} \cdot H_{\text{Na}} \cdot (V - V_{\text{Na}} ) $$

(13)

where g _Na is the maximal conductance of the Na⁺ current, M _Na and H _Na are the activation and inactivation constants and V _Na is the reversal potential of the Na⁺ current. The activation and inactivation constants at the soma are given by

$$ \begin{aligned} M_{\text{Na}} & = \alpha_{\text{M}} (V)/(\alpha_{\text{M}} (V) + \beta {}_{\text{M}}(V)) \\ \alpha_{\text{M}} (V) & = 0.32 \cdot ( - 46.9 - V)/(\exp (( - 46.9 - V)/4.0) - 1.0) \\ \beta_{\text{M}} (V) & = 0.28 \cdot (V + 19.9)/(\exp ((V + 19.9)/5.0) - 1.0) \\ H_{\text{Na}}^{\prime } & = \alpha_{\text{H}} (V) - (\alpha_{\text{H}} (V) + \beta_{\text{H}} (V)) \cdot H_{\text{Na}} \\ \alpha_{\text{H}} (V) & = 0.128 \cdot \exp (( - 43 - V)/18) \\ \beta_{\text{H}} (V) & = 4/(1 + \exp (( - 20 - V)/5) \\ \end{aligned} $$

The sodium current at the dendrite is described by

$$ I_{\text{Na,d}} = - g_{\text{Na,d}} \cdot M_{\text{Na,d}}^{2} \cdot H_{\text{Na,d}} \cdot D_{\text{Na,d}} \cdot (V_{\text{d}} - V_{\text{Na}} ) $$

(14)

where

$$ \begin{aligned} M_{\text{Na,d}}^{\prime } & = \left( {M_{{\infty_{\text{Na,d}} }} - M_{\text{Na,d}} } \right)/\tau_{{{\text{M}}_{\text{Na,d}} }} \\ M_{{\infty_{\text{Na,d}} }} & = 1/(1 + \exp (( - V_{\text{d}} - 40)/3)) \\ \tau_{{{\text{M}}_{\text{Na,d}} }} & = 0.1 \\ H_{\text{Na,d}}^{\prime } & = \left( {H_{{\infty_{\text{Na,d}} }} - H_{\text{Na,d}} } \right)/\tau_{{{\text{H}}_{\text{Na,d}} }} \\ H_{{\infty_{\text{Na,d}} }} & = 1/(1 + \exp ((V_{\text{d}} + 45)/3)) \\ \tau_{{{\text{H}}_{\text{Na,d}} }} & = 0.5 \\ D_{\text{Na,d}}^{\prime } & = \left( {D_{{\infty_{\text{Na,d}} }} - D_{\text{Na,d}} } \right)/\tau_{{{\text{D}}_{\text{Na,d}} }} \\ D_{{\infty_{\text{Na,d}} }} & = (1 + {\text{natt}} \cdot \exp ((V_{\text{d}} + 60)/2))/(1 + \exp ((V_{\text{d}} + 60)/2)) \\ \tau_{{{\text{D}}_{\text{Na,d}} }} & = \max (0.1,(0.00333 \cdot \exp (0.0024 \cdot (V_{\text{d}} + 60) \cdot Q))/(1 + \exp (0.0012 \cdot (V_{\text{d}} + 60) \cdot Q))) \\ Q & = 96,480/(8.315 \cdot (273.16^{^\circ } + T)) \\ \end{aligned} $$

where T is the temperature in Celcius and natt is the Na⁺ attenuation. The type-A K⁺ current at the soma and dendrite is given by

$$ I_{{{\text{K}}_{\text{A,d}} }} = - g_{{{\text{K}}_{\text{A,d}} }} \cdot A_{\text{d}} \cdot B_{\text{d}} \cdot (V_{\text{d}} - V_{\text{K}} ) $$

(15)

The activation and inactivation constants are given by

$$ \begin{aligned} A_{\text{d}}^{\prime } & = \left( {A_{{\infty_{\text{d}} }} - A_{\text{d}} } \right)/\tau_{{A_{\text{d}} }} \\ A_{{\infty_{\text{d}} }} & = 1/(1 + A_{{\alpha ,{\text{d}}}} ) \\ A_{{\alpha ,{\text{d}}}} & = \exp ({\text{asap}} \cdot \varsigma (V_{\text{d}} ) \cdot (V_{\text{d}} + 1) \cdot Q) \\ A_{{\beta ,{\text{d}}}} & = \exp (0.00039 \cdot Q \cdot (V_{\text{d}} + 1) \cdot \varsigma_{2} (V_{\text{d}} )) \\ \tau_{{A_{\text{d}} }} & = \max (A_{{\beta ,{\text{d}}}} /((1 + A_{{\alpha ,{\text{d}}}} ) \cdot QT \cdot 0.1),0.1) \\ \varsigma (V_{\text{d}} ) & = - 1.5 - (1/(1 + \exp ((V_{\text{d}} + \varsigma_{\text{p}} )/5))) \\ \varsigma_{2} (V_{\text{d}} ) & = - 1.8 - (1/(1 + \exp ((V_{\text{d}} + 40)/5))) \\ B_{\text{d}}^{\prime } & = (B_{{\infty_{\text{d}} }} - B_{\text{d}} )/\tau_{{B_{\text{d}} }} \\ B_{{\infty_{\text{d}} }} & = 0.3 + 0.7/(1 + \exp ({\text{inact}}_{2} \cdot (V_{\text{s}} + {\text{inact}}) \cdot Q)) \\ \tau_{{B_{\text{d}} }} & = \kappa \cdot \max ({\text{inact}}_{3} \cdot (V_{\text{s}} + {\text{inact}}_{4} ),{\text{inact}}_{5} ) \\ \end{aligned} $$

The delayed rectifier K⁺ current at the axon and soma is given by

$$ I_{{{\text{K}}_{\text{dr}} }} = - g_{{{\text{K}}_{\text{dr}} }} \cdot N \cdot (V - V_{\text{K}} ) $$

(16)

where g _Kds is the maximal conductance. The activation constant N is given by

$$ \begin{aligned} N^{\prime } & = \alpha_{N} (V) - (\alpha_{N} (V) + \beta_{N} (V)) \cdot N \\ \alpha_{N} (V) & = 0.016 \cdot ( - 24.9 - V)/(\exp (( - 24.9 - V)/5) - 1) \\ \beta_{N} (V) & = 0.25 \cdot \exp ( - 1 - 0.025 \cdot V) \\ \end{aligned} $$

The delayed rectifier K⁺ current at the dendrite is given by

$$ I_{{{\text{K}}_{\text{dr,d}} }} = - g_{{{\text{K}}_{\text{dr,d}} }} \cdot N_{\text{d}}^{2} \cdot (V_{\text{d}} - V_{\text{K}} ) $$

(17)

where g _Kdr,d is the maximal conductance. The activation constant N _d is given by

$$ \begin{aligned} N_{\text{d}}^{\prime } & = (N_{{\infty_{\text{d}} }} - N_{\text{d}} )/\tau_{{N_{\text{d}} }} \\ N_{{\infty_{\text{d}} }} & = 1/(1 + \exp (( - V_{\text{d}} - 42)/2) \\ \tau_{{N_{\text{d}} }} & = 2.2 \\ \end{aligned} $$

The medium Ca²⁺-activated K⁺ after-hyperpolarization current at the soma is given by

$$ I_{\text{mAHP}} = - g_{\text{mAHP}} \cdot Q_{\text{m}} \cdot (V_{\text{s}} - V_{\text{K}} ) $$

(18)

where g _KmAHP is the maximal conductance. The activation constant Q _m is given by

$$ \begin{aligned} Q_{\text{m}}^{\prime } & = \left( {Q_{{{\text{m}}_{\infty } }} - Q_{\text{m}} } \right)/\tau_{{Q_{\text{m}} }} \\ Q_{{{\text{m}}_{\infty } }} & = {\text{qhat}} \cdot Q_{{{\text{m}}_{{{\upalpha}}} }} \cdot \tau_{{Q_{\text{m}} }} \\ Q_{{{\text{m}}_{{{\upalpha}}} }} & = {\text{qma}} \cdot \chi_{\text{s}} /(0.001 \cdot \chi_{\text{s}} + 0.18 \cdot \exp ( - 1.68 \cdot V_{\text{s}} \cdot Q)) \\ Q_{{{\text{m}}_{{{\upbeta}}} }} & = ({\text{qmb}} \cdot \exp ( - 0.022 \cdot V_{\text{s}} \cdot Q))/(\exp ( - 0.022 \cdot V_{\text{s}} \cdot Q) + 0.001 \cdot \chi_{\text{s}} ) \\ \tau_{{Q_{\text{m}} }} & = 1/\left( {Q_{{{\text{m}}_{{{\upalpha}}} }} + Q_{{{\text{m}}_{{{\upbeta}}} }} } \right) \\ \end{aligned} $$

The h-current [13, 14] at the soma and dendrite is described by

$$ I_{\text{h}} = - g_{\text{h}} \cdot tt \cdot (V - E_{\text{h}} ) $$

(19)

$$ \begin{aligned} \frac{{{\text{d}}tt}}{{{\text{d}}t}} & = \frac{{tt_{\infty } - tt}}{{\tau_{tt} }} \\ tt_{\infty } & = \frac{1}{{1 + {\text{e}}^{{ - (V - V_{\text{half}} )/k_{l} }} }}\quad \tau_{tt} = \frac{{{\text{e}}^{{0.0378 \cdot \varsigma \cdot {\text{gmt}} \cdot (V - V_{\text{halft}} )}} }}{{qtl \cdot q10^{(T - 33)/10} \cdot a0t \cdot (1 + a_{tt} )}} \\ a_{tt} & = {\text{e}}^{{0.00378 \cdot \varsigma \cdot (V - V_{\text{halft}} )}} \\ \end{aligned} $$

where g_h is the maximal conductance of the h-current and E _h is the reversal potential. The L-type Ca²⁺ current at the soma is described by

$$ I_{{{\text{CaL}}_{\text{s}} }} = - g_{{{\text{CaL}}_{\text{s}} }} \cdot S_{\text{s}} \cdot g_{\text{hk}} (V_{\text{s}} ,\chi_{\text{s}} ) \cdot (1/(1 + \chi_{\text{s}} )) $$

(20)

where g _CaL,s is the maximal conductance and

$$ \begin{aligned} S_{\text{s}}^{\prime } & = \left( {S_{{\infty_{\text{s}} }} - S_{\text{s}} } \right)/\tau_{{s_{\text{s}} }} \\ S_{{\infty_{\text{s}} }} & = \alpha_{\text{s}} (V_{\text{s}} )/(\alpha_{\text{a}} (V_{\text{s}} ) + \beta_{\text{s}} (V_{\text{s}} )) \\ \tau_{{S_{\text{s}} }} & = 1/(5 \cdot (\alpha_{\text{s}} (V_{\text{s}} ) + \beta_{\text{s}} (V_{\text{s}} ))) \\ \alpha_{\text{s}} (V_{\text{s}} ) & = - 0.055 \cdot (V_{\text{s}} + 27.01)/(\exp (( - V_{\text{s}} - 27.01)/3.8) - 1) \\ \beta_{\text{s}} (V_{\text{s}} ) & = 0.94 \cdot \exp (( - V_{\text{s}} - 63.01)/17) \\ xx & = 0.0853 \cdot (273.16 + T)/2 \\ f(z) & = (1 - z/2) \cdot f_{2} (z) + (z/(\exp (z) - 1)) \cdot f_{3} (z) \\ f_{2} (z) & = H(0.0001 - \left| z \right|) \\ f_{3} (z) & = H(\left| z \right| - 0.0001) \\ g_{\text{hk}} & = - xx \cdot (1 - ((\chi_{\text{s}} /{\text{Ca}}) \cdot \exp (V_{\text{s}} /xx))) \cdot f(V_{\text{s}} /xx) \\ \end{aligned} $$

The Ca²⁺ concentrations in the soma and dendrites [71] are given by

$$ \chi_{\text{s}}^{\prime } = \phi_{\text{s}} \cdot I_{{{\text{CaL}}_{\text{s}} }} - (\beta_{\text{s}} \cdot (\chi_{\text{s}} - \chi_{{0,{\text{s}}}} )) + (\chi_{\text{pd}} - \chi_{\text{s}} )/{\text{Ca}}_{\tau } - (\beta_{\text{s}} /{\text{nonc}}) \cdot \chi_{\text{s}}^{2} $$

(21)

$$ \chi_{\text{pd}}^{\prime } = \phi_{\text{d}} \cdot (I_{{{\text{CaL}}_{\text{d}} }} + I_{\text{Ca,NMDA}} ) - \beta_{\text{d}} \cdot (\chi_{\text{pd}} - \chi_{{0,{\text{d}}}} ) - (\beta_{\text{d}} /{\text{nonc}}) \cdot \chi_{\text{pd}}^{2} - {\text{buff}} \cdot \chi_{\text{pd}} $$

(22)

$$ \chi_{\text{dd}}^{\prime } = \phi_{\text{d}} \cdot \left( {I_{{{\text{CaL}}_{\text{d}} }} + I_{\text{Ca,NMDA}} } \right) - \beta_{\text{d}} \cdot (\chi_{\text{dd}} - \chi_{{0,{\text{d}}}} ) - (\beta_{\text{d}} /{\text{nonc}}) \cdot \chi_{\text{dd}}^{2} - {\text{buff}} \cdot \chi_{\text{dd}} $$

(23)

The L-type Ca²⁺ current at the dendrite is described by

$$ I_{{{\text{CaL}}_{\text{d}} }} = - g_{{{\text{CaL}}_{\text{d}} }} \cdot S_{\text{d}}^{3} \cdot T_{\text{d}} \cdot (V_{\text{d}} - V_{\text{Ca}} ) $$

(24)

$$ \begin{aligned} S_{\text{d}}^{\prime } & = \left( {S_{{\infty_{\text{d}} }} - S_{\text{d}} } \right)/\tau_{{{\text{s}}_{\text{d}} }} \\ S_{{\infty_{\text{d}} }} & = 1/(1 + \exp ( - V_{\text{d}} - 37)) \\ \tau_{{{\text{s}}_{\text{d}} }} & = s_{3} + s_{1} /(1 + \exp (V_{\text{d}} + s_{2} )) \\ T_{\text{d}}^{\prime } & = (T_{{\infty_{\text{d}} }} - T_{\text{d}} )/\tau_{{T_{\text{d}} }} \\ T_{{\infty_{\text{d}} }} & = 1/(1 + \exp ((V_{\text{d}} + 41)/0.5)) \\ \tau_{{T_{\text{d}} }} & = 29 \\ \end{aligned} $$

The calcium detector model is governed by the following six equations:

$$ P^{\prime } = (\phi_{\text{a}} (\chi_{\text{d}} ) - c_{\text{p}} \cdot A \cdot P)/\tau_{\text{p}} $$

(25)

$$ V^{\prime } = (\phi_{\text{b}} (\chi_{\text{d}} ) - V)/\tau_{V} $$

(26)

$$ A^{\prime } = (\phi_{\text{c}} (\chi_{\text{d}} ) - A)/\tau_{A} $$

(27)

$$ B^{\prime } = (\phi_{\text{e}} (A) - B - c_{\text{d}} \cdot B \cdot V)/\tau_{B} $$

(28)

$$ D^{\prime } = (\phi_{\text{d}} (B) - D)/\tau_{D} $$

(29)

$$ W^{\prime } = (\alpha_{\text{w}} /(1 + \exp ((P - a)/p_{a} )) - \beta_{w} /(1 + (\exp ((D - d)/p_{d} )) - W)/\tau_{w} $$

(30)

where P is the potentiation detector dynamics, V is the veto detector dynamics, D is the depression detector dynamics, A and B are the intermediate steps leading up to D and W is the readout variable (see Fig. 2). The Hill equations are

$$ \begin{aligned} \phi_{\text{a}} (x) & = {\text{num}}_{\text{a}} \cdot ((x/{\text{CmHC}})^{\text{CmHN}} )/(1 + (x/{\text{CmHC}})^{\text{CmHN}} ) \\ \phi_{\text{b}} (x) & = {\text{num}}_{\text{b}} \cdot ((x/{\text{CnHC}})^{\text{CnHN}} )/(1 + (x/{\text{CnHC}})^{\text{CnHN}} ) \\ \phi_{\text{c}} (x) & = {\text{num}}_{\text{c}} /(1 + \exp ((x - \theta_{\text{c}} )/\sigma_{\text{c}} )) \\ \phi_{\text{d}} (x) & = {\text{num}}_{\text{d}} /(1 + \exp ((x - \theta_{\text{d}} )/\sigma_{\text{d}} )) \\ \phi_{\text{e}} (x) & = {\text{num}}_{\text{e}} /(1 + \exp ((x - \theta_{\text{e}} )/\sigma_{\text{e}} )) \\ \end{aligned} $$

The calcium detector parameter values are displayed in Table 2.

Table 2 Calcium detector model parameter values

Basket, Axoaxonic and Bistratified Cells

$$ C_{\text{m}} \frac{{{\text{d}}V}}{{{\text{d}}t}} = I_{\text{L}} + I_{\text{Na}} + I_{{{\text{K}}_{\text{dr}} }} + I_{\text{A}} + I_{\text{in}} + I_{\text{syn}} $$

(31)

where C _m is the membrane capacitance, V is the membrane potential, I _L is the leak current, I _Na is the sodium current, I _Kdr is the fast delayed rectifier K⁺ current, I _A is the A-type K⁺ current and I _syn is the synaptic current.

The sodium current and its kinetics are described by

$$ I_{\text{Na}} = g_{\text{Na}} m^{3} h(V - E_{\text{Na}} ) $$

(32)

$$ \begin{aligned} \frac{{{\text{d}}m}}{{{\text{d}}t}} & = \alpha_{m} (1 - m) - \beta_{m} m,\quad \alpha_{m} = \frac{0.1(V + 40)}{{(1 - {\text{e}}^{(V + 40)/10} )}},\quad \beta_{\text{m}} = 4 \cdot {\text{e}}^{( - (v + 65)/18)} \\ \frac{{{\text{d}}h}}{{{\text{d}}t}} & = \alpha_{h} (1 - h) - \beta_{h} h,\quad \alpha_{h} = 0.07 \cdot {\text{e}}^{ - (V + 65)/20} ,\quad \beta_{\text{h}} = \frac{1}{{(1 + {\text{e}}^{ - (V + 35)/10} )}}. \\ \end{aligned} $$

The fast delayed rectifier K⁺ current I _Kdr is given by

$$ I_{\text{Kdr}} = g_{\text{Kdr}} n^{4} (V - E_{\text{K}} ) $$

(33)

$$ \frac{{{\text{d}}n}}{{{\text{d}}t}} = \alpha_{n} (1 - n) - \beta_{n} n,\quad \alpha_{n} = \frac{0.01(V + 55)}{{(1 - {\text{e}}^{ - (V + 55)/10} )}},\quad \beta_{n} = 0.125{\text{e}}^{ - (v + 65)/80} . $$

The A-type K⁺ current I _A is described by

$$ I_{\text{A}} = g_{\text{A}} ab(V - E_{\text{k}} ) $$

(34)

$$ \begin{aligned} \frac{{{\text{d}}a}}{{{\text{d}}t}} & = \alpha_{a} (1 - a) - \beta_{a} a,\quad \alpha_{a} = \frac{0.02(13.1 - V)}{{{\text{e}}^{{\left( {\frac{13.1 - V}{10}} \right)}} - 1}},\quad \beta_{a} = \frac{0.0175(V - 40.1)}{{{\text{e}}^{{\left( {\frac{V - 40.1}{10}} \right)}} - 1}} \\ \frac{{{\text{d}}b}}{{{\text{d}}t}} & = \alpha_{b} (1 - b) - \beta_{b} b,\quad \alpha_{b} = 0.0016\,{\text{e}}^{{\left( {\frac{ - 13 - V}{18}} \right)}} ,\quad \beta_{b} = \frac{0.05}{{1 + {\text{e}}^{{\left( {\frac{10.1 - V}{5}} \right)}} }} \\ \end{aligned} $$

The ionic parameter values are depicted in Table 3.

Table 3 Inhibitory cell parameter values

OLM Cell

$$ C_{\text{m}} \frac{{{\text{d}}V}}{{{\text{d}}t}} = I_{\text{L}} + I_{\text{Na}} + I_{{{\text{K}}_{\text{dr}} }} + I_{\text{NaP}} + I_{\text{H}} + I_{\text{syn}} + I_{\text{in}} $$

(35)

where C _m is the membrane capacitance, V is the membrane potential, I _L is the leak current, I _Na is the sodium current, I _Kdr is the fast delayed rectifier K⁺ current, I _NaP is the persistent sodium current, I _h is the h-current and I _syn is the synaptic current.

The sodium current and its kinetics are described by

$$ I_{\text{Na}} = g_{\text{Na}} m^{3} h(V - E_{\text{Na}} ) $$

(36)

$$ \begin{aligned} \frac{{{\text{d}}m}}{{{\text{d}}t}} & = \alpha_{m} (1 - m) - \beta_{m} m,\quad \alpha_{m} = \frac{0.1(V + 40)}{{(1 - {\text{e}}^{(V + 40)/10} )}},\quad \beta_{m} = 4 \cdot {\text{e}}^{( - (v + 65)/18)} \\ \frac{{{\text{d}}h}}{{{\text{d}}t}} & = \alpha_{h} (1 - h) - \beta_{h} h,\quad \alpha_{h} = 0.07 \cdot {\text{e}}^{ - (V + 65)/20} ,\quad \beta_{h} = \frac{1}{{(1 + {\text{e}}^{ - (V + 35)/10} )}}. \\ \end{aligned} $$

The fast delayed rectifier K⁺ current I _Kdr is given by

$$ I_{\text{Kdr}} = g_{\text{Kdr}} n^{4} (V - E_{\text{K}} ) $$

(37)

$$ \frac{{{\text{d}}n}}{{{\text{d}}t}} = \alpha_{n} (1 - n) - \beta_{n} n,\quad \alpha_{n} = \frac{0.01(V + 55)}{{(1 - {\text{e}}^{ - (V + 55)/10} )}},\quad \beta_{n} = 0.125{\text{e}}^{ - (v + 65)/80} $$

The NaP current was assembled from the Kunec et al.’s [49] and Dickson et al.’s [19, 25, 26, 76] studies, and it was described by

$$ I_{\text{NaP}} = - g_{\text{NaP}} \cdot m_{\text{po}} \cdot (V - V_{\text{Na}} ) $$

(38)

$$ \begin{aligned} \frac{{{\text{d}}m_{\text{po}} }}{{{\text{d}}t}} & = \alpha_{{m_{\text{po}} }} (V)(1 - m_{\text{po}} ) - \beta_{{m_{\text{po}} }} (V) \cdot m_{\text{po}} \\ \alpha_{{m_{\text{po}} }} & = \frac{1}{{0.15(1 + {\text{e}}^{ - (V + 38)/6.5} )}},\quad \beta_{{m_{\text{po}} }} = \frac{{{\text{e}}^{ - (V + 38)/6.5} }}{{0.15(1 + {\text{e}}^{ - (V + 38)/6.5} )}} \\ \end{aligned} $$

Similarly, the h-current was assembled from Kunec et al.’s [49] and Dickson et al.’s [19, 25, 26, 76] studies, and it was described by

$$ I_{\text{h}} = - g_{\text{h}} (0.65\lambda_{\text{fo}} + 0.35\lambda_{\text{so}} )(V - V_{\text{h}} ) $$

(39)

$$ \begin{aligned} \frac{{{\text{d}}\lambda_{\text{fo}} }}{{{\text{d}}t}} & = \frac{{\lambda_{{{\text{f}}\infty }} (V) - \lambda_{\text{fo}} }}{{\tau_{{\lambda {\text{f}}}} (V)}},\quad \lambda_{{{\text{f}}\infty }} (V) = \frac{1}{{(1 + {\text{e}}^{(V + 79.2)/9.78} )}},\quad \tau_{{\lambda {\text{f}}}} (V) = \frac{0.51}{{{\text{e}}^{(v - 1.7)/10} + {\text{e}}^{ - (V + 340)/52} }} + 1 \\ \frac{{{\text{d}}\lambda_{\text{so}} }}{{{\text{d}}t}} & = \frac{{\lambda_{{{\text{s}}\infty }} (V) - \lambda_{\text{so}} }}{{\tau_{{\lambda {\text{s}}}} (V)}},\quad \lambda_{{{\text{s}}\infty }} (V) = \frac{1}{{(1 + {\text{e}}^{(V + 2.83)/15.9} )^{58} }},\quad \tau_{{\lambda {\text{s}}}} (V) = \frac{5.6}{{{\text{e}}^{(v - 1.7)/14} + {\text{e}}^{ - (V + 260)/43} }} + 1. \\ \end{aligned} $$

The ionic parameter values are depicted in Table 3.

Input-to-Cell Synaptic Currents

The Ca²⁺-NMDA, AMPA, GABA_A and NMDA synaptic currents are given by [66] and references therein

$$ I_{\text{Ca,NMDA}} = - g_{\text{syn}} \cdot s_{\text{NMDA}} \cdot m_{\text{Ca,NMDA}} \cdot (V_{\text{d}} - V_{\text{Ca,NMDA}} ) $$

(40)

$$ I_{\text{NMDA}} = - g_{\text{syn}} \cdot s_{\text{NMDA}} \cdot m_{\text{NMDA}} \cdot (V_{\text{d}} - V_{\text{NMDA}} ) $$

(41)

$$ I_{\text{AMPA}} = - g_{\text{syn}} \cdot s_{\text{AMPA}} \cdot (V_{\text{d}} - V_{\text{AMPA}} ) $$

(42)

$$ I_{\text{GABA}} = - g_{\text{syn}} \cdot s_{\text{GABA}} \cdot (V_{\text{d}} - V_{\text{GABA}} ) $$

(43)

where g _syn is the synaptic conductance expressed either by Eqs. 47 or 1–3 and

$$ \begin{aligned} m_{\text{NMDA}} & = 1/(1 + 0.3 \cdot {\text{Mg}} \cdot \exp ( - 0.062 \cdot V_{\text{d}} )) \\ m_{\text{Ca,NMDA}} & = 1/(1 + 0.3 \cdot {\text{Mg}} \cdot \exp ( - 0.124 \cdot V_{\text{d}} )) \\ \end{aligned} $$

with Mg²⁺ = 2 mM. The activation equations for AMPA, NMDA and GABA_A currents are

$$ s_{x} = s_{{x_{\text{fast}} }} + s_{{x_{\text{slow}} }} + s_{{x_{\text{rise}} }} $$

(44)

where x stands for AMPA, NMDA, GABA and

$$ \begin{aligned} s_{{{\text{NMDA}}_{\text{rise}} }}^{\prime } & = - 20 \cdot (1 - s_{{{\text{NMDA}}_{\text{fast}} }} - s_{{{\text{NMDA}}_{\text{slow}} }} ) \cdot F_{\text{pre}} - (1/2) \cdot s_{{{\text{NMDA}}_{\text{rise}} }} \\ s_{{{\text{NMDA}}_{\text{fast}} }}^{\prime } & = 20 \cdot (0.527 - s_{{{\text{NMDA}}_{\text{fast}} }} ) \cdot F_{\text{pre}} - (1/10) \cdot s_{{{\text{NMDA}}_{\text{fast}} }} \\ s_{{{\text{NMDA}}_{\text{slow}} }}^{\prime } & = 20 \cdot (0.473 - s_{{{\text{NMDA}}_{\text{slow}} }} ) \cdot F_{\text{pre}} - (1/45) \cdot s_{{{\text{NMDA}}_{\text{slow}} }} , \\ \end{aligned} $$

$$ \begin{aligned} s_{{{\text{AMPA}}_{\text{rise}} }}^{\prime } & = - 20 \cdot (1 - s_{{{\text{AMPA}}_{\text{fast}} }} - s_{{{\text{AMPA}}_{\text{slow}} }} ) \cdot F_{\text{pre}} - (1/0.58) \cdot s_{{{\text{AMPA}}_{\text{rise}} }} \\ s_{{{\text{AMPA}}_{\text{fast}} }}^{\prime } & = 20 \cdot (0.903 - s_{{{\text{AMPA}}_{\text{fast}} }} ) \cdot F_{\text{pre}} - (1/7.6) \cdot s_{{{\text{AMPA}}_{\text{fast}} }} \\ s_{{{\text{AMPA}}_{\text{slow}} }}^{\prime } & = 20 \cdot (0.097 - s_{{{\text{AMPA}}_{\text{slow}} }} ) \cdot F_{\text{pre}} - (1/25.69) \cdot s_{{{\text{AMPA}}_{\text{slow}} }} \\ \end{aligned} $$

and

$$ \begin{aligned} s_{{{\text{GABA}}_{\text{rise}} }}^{\prime } & = - 20 \cdot (1 - s_{{{\text{GABA}}_{\text{fast}} }} - s_{{{\text{GABA}}_{\text{slow}} }} ) \cdot F_{\text{pre}} - (1/1.18) \cdot s_{{{\text{GABA}}_{\text{rise}} }} \\ s_{{{\text{GABA}}_{\text{fast}} }}^{\prime } & = 20 \cdot (0.803 - s_{{{\text{GABA}}_{\text{fast}} }} ) \cdot F_{\text{pre}} - (1/8.5) \cdot s_{{{\text{GABA}}_{\text{fast}} }} \\ s_{{{\text{GABA}}_{\text{slow}} }}^{\prime } & = 20 \cdot (0.197 - s_{{{\text{GABA}}_{\text{slow}} }} ) \cdot F_{\text{pre}} - (1/30.01) \cdot s_{{{\text{GABA}}_{\text{slow}} }} \\ \end{aligned} $$

where F _pre is the input spike generator simulating the CA3 Schaffer collateral, the EC perforant path and the MS inputs. The input-to-cell synaptic parameter values are displayed in Table 4.

Table 4 Input-to-cell and cell-to-cell synaptic parameter values

Table 5 Cell-to-cell synaptic parameter values

Input Spike Generator

The input spike generator simulating the CA3 Schaffer collateral, the EC perforant path and the MS inputs were described by

$$ F_{\text{pre}} = H(t - 1) \cdot (H(\sin (2\pi \cdot (t - 2)/T)) \cdot (1 - H(\sin (2\pi \cdot (t - 1)/T)))) $$

(45)

where T is the period of oscillation and H() is the Heaviside function.

Cell-to-Cell Synaptic Currents

The synaptic current is given by

$$ I_{\text{syn}} = g_{\text{syn}} \cdot s \cdot (V - E_{\text{rev}} ) $$

(46)

where g _syn is the synaptic conductance and E _rev is the reversal potential. The synaptic conductance is expressed by

$$ g_{\text{syn}} = w \cdot g_{\max } $$

(47)

where g _max is the maximal synaptic conductance and w is the synaptic strength. The values of the synaptic strengths are given in Table 6. In the model, three synaptic currents are included: AMPA, NMDA and GABA_A. The values of the synaptic parameters are displayed in Table 4. The gating variable, s, which represents the fraction of the open synaptic ion channels, obeys the following differential equation

$$ \frac{{{\text{d}}s}}{{{\text{d}}t}} = \alpha \cdot F(V_{\text{pre}} ) \cdot (1 - s) - \beta \cdot s $$

(48)

where the normalized concentration of the postsynaptic transmitter–receptor complex, F(V_pre), is assumed to be an instantaneous and sigmoid functions of the presynaptic membrane potential

$$ F(V_{\text{pre}} ) = 1/\left( {1 + {\text{e}}^{{ - (V_{\text{pre}} - \theta )/2}} } \right) $$

(49)

where θ = 0 mV is high enough so that the transmitter release occurs only when the presynaptic cell emits a spike [16]. The values of the channel opening and closing rates are displayed in Table 5.

Table 6 Synaptic strength parameter values