Dynamics analysis of the hippocampal neuronal model subjected to cholinergic action related with Alzheimer’s disease

Abstract

There are evidences that the region of hippocampus is affected in the early stage of Alzheimer’s disease (AD). Moreover, the hippocampal pyramidal neurons receive cholinergic input from the medial septum. Thus, this study, based on the results of electrophysiological experiments, first constructs a modified hippocampal CA1 pyramidal neuronal model by introducing two new currents of M-current and calcium ion-activated potassium ion current to depict the cholinergic input receiving from the medial septum, and then explores how acetylcholine deficiency and beta-amyloid accumulation under the pathological condition of AD influence the neuronal dynamics in terms of theta band power and spiking frequency using computational approach. By simulating acetylcholine potentiated M-current and calcium ion-activated potassium ion current, numerical results reveal that the relative theta band power increases significantly and the firing rate decreases obviously when acetylcholine is deficient. Similarly, by simulating beta-amyloid enhanced delay rectification potassium ion current, we also detect that the relative theta band power increases as well as the firing rate decreases remarkably as beta-amyloid is accumulated. In addition, the mechanism underlying these dynamical changes in theta rhythm and firing behavior is investigated by nonlinear behavioral analysis, which demonstrates that both deficiency in acetylcholine and accumulation in beta-amyloid can promote the emergence of stable equilibrium state in this modified hippocampal neuronal model. Note that acetylcholine deficiency together with beta-amyloid deposition plays key role in the pathogenesis of AD. We expect these findings could have important implications on better understanding pathogenesis and expounding potential biomarkers for AD.

Appendix

The present modified hippocampal pyramidal neuronal mathematical model is described by the following differential equation:

$$ C\frac{dV}{dt} = - I_{L} - I_{Na} - I_{K} - I_{M} - I_{AHP} - I_{A} + I $$

(2)

where $C$ is the membrane capacitance, $V$ is the membrane potential, $I_{L}$ is the leakage current, $I_{Na}$ is the transient $Na^{ + }$ current, $I_{K}$ is the delay rectification $K^{ + }$ current, $I_{A}$ is the A-type instantaneous $K^{ + }$ current, $I_{M}$ is the muscarine-sensitive $K^{ + }$ current, $I_{AHP}$ is the calcium ion-activated potassium ion current and $I$ is the stimulation current.

All of the above ionic currents are modelled by the Hodgkin–Huxley type, thus the gating variable $x$ satisfies the following first-order kinetics ($x$ can be $h$, $n$, $z$, $r$ and $b$):

$$ \frac{dx}{dt} = \varphi_{x} \frac{{x_{\infty } (V) - x}}{{\tau_{x} }}. $$

(3)

The model (2) has eight variables, which are the membrane potential variable $V$, transient $Na^{ + }$ current inactivation variable $h$, delayed rectified $K^{ + }$ current activation variable $n$, A-type instantaneous $K^{ + }$ current inactivation variable $b$, muscarine-sensitive $K^{ + }$ current activated variable $z$, high-threshold $Ca^{2 + }$ current-activated variable $r$, calcium ion-activated potassium ion current-activated variable $q$ and intramembrane calcium ion concentration variable $[Ca^{2 + } ]_{i}$. The activation gate variables $m$ and $a$ are replaced by activation curves $m_{\infty } (V)$ and $a_{\infty } (V)$, respectively. $h_{\infty }$, $r_{\infty }$ and $b_{\infty }$ stand for activation curves of activation gate variables $h$, $r$ and $b$, respectively. $n_{\infty }$, $z_{\infty }$ and $q_{\infty }$ stand for inactivation curves of inactivation gate variables $n$, $z$ and $q$, respectively. For numerical simulation the parameters are selected as follows: $C = 1\upmu {{\rm F/cm}}^{2}$, $g_{L} = 0.05\,{{\rm mS/cm}}^{2}$, $E_{L} = - 60\,{{\rm mV}}$, $\varphi = 1$, $g_{Na} = 35\,{{\rm mS/cm}}^{2}$, $g_{K} = 10\,{{\rm mS/cm}}^{2}$, $g_{M} = 0.1\,{{\rm mS/cm}}^{2}$, $g_{AHP} = 0.15\,{{\rm mS/cm}}^{2}$, $g_{A} = 1\,{{\rm mS/cm}}^{2}$; $E_{Na} = 55\,{{\rm mV}}$, $E_{K} = - 90\,{{\rm mV}}$. Without loss of generality, in the modified model (2) the state variable $V$, $h$, $n$, $z$, $r$, $[Ca^{2 + } ]_{i}$, $q$ and $b$ are initially set as $V = - 65\,{{\rm mV}}$, $h = 0.1$, $n = 0.1$, $z = 0.1$, $r = 0.1$, $[Ca^{2 + } ]_{i} = 0.05$, $q = 0.1$, $b = 0.1$.