Beta-amyloid induced changes in A-type K⁺ current can alter hippocampo-septal network dynamics

Abstract

Alzheimer’s disease (AD) progression is usually associated with memory deficits and cognitive decline. A hallmark of AD is the accumulation of beta-amyloid (Aβ) peptide, which is known to affect the hippocampal pyramidal neurons in the early stage of AD. Previous studies have shown that Aβ can block A-type K⁺ currents in the hippocampal pyramidal neurons and enhance the neuronal excitability. However, the mechanisms underlying such changes and the effects of the hyper-excited pyramidal neurons on the hippocampo-septal network dynamics are still to be investigated. In this paper, Aβ-blocked A-type current is simulated, and the resulting neuronal and network dynamical changes are evaluated in terms of the theta band power. The simulation results demonstrate an initial slight but significant theta band power increase as the A-type current starts to decrease. However, the theta band power eventually decreases as the A-type current is further decreased. Our analysis demonstrates that Aβ blocked A-type currents can increase the pyramidal neuronal excitability by preventing the emergence of a steady state. The increased theta band power is due to more pyramidal neurons recruited into spiking mode during the peak of pyramidal theta oscillations. However, the decreased theta band power is caused by the spiking phase relationship between different neuronal populations, which is critical for theta oscillation, is violated by the hyper-excited pyramidal neurons. Our findings could provide potential implications on some AD symptoms, such as memory deficits and AD caused epilepsy.

Appendix

The membrane capacitance C = 1 μF/cm ² for all of the follow equations, therefore it will be ignored. The τ in (ms); E and V in (mV); I in (μA/cm ²); g in (mS/cm ²); α and β in (ms ⁻¹); K in (μM); B in (μM(msμA)⁻¹ cm ²) and the rest are dimensionless constant. We used an Euler method for numerically integrating the stochastic differential equations, using a time step of 0.01 ms. Smaller time steps do not change our results.

1.1 Neuronal dynamics

The pyramidal somatic and dendritic membrane potentials, denoted by V_s and V_d, obtains the following equations:

$$ {\dot{V}_s} = - {I_L} - {I_{{Na}}} - {I_K} - {I_{{Ca}}} - {I_A} - {I_{{CT}}} - \frac{{{g_c}}}{p}\left( {{V_s} - {V_d}} \right) - {I_{{syn,s}}} + I {\dot{V}_s} = - {I_L} - {I_{{Na}}} - {I_K} - {I_{{Ca}}} - {I_A} - {I_{{CT}}} - \frac{{{g_c}}}{p}\left( {{V_s} - {V_d}} \right) - {I_{{syn,s}}} + I $$

(A1)

$$ {\dot{V}_d} = - {I_L} - {I_{{Ca}}} - {I_{{AHP}}} - {I_A} - {I_{{CT}}} - \frac{{{g_c}}}{{1 - p}}\left( {{V_d} - {V_s}} \right) - {I_{{syn,d}}} {\dot{V}_d} = - {I_L} - {I_{{Ca}}} - {I_{{AHP}}} - {I_A} - {I_{{CT}}} - \frac{{{g_c}}}{{1 - p}}\left( {{V_d} - {V_s}} \right) - {I_{{syn,d}}} $$

(A2)

Where g _c  = 2 mS/cm ² is the coupling conductance between soma and dendrite, p=somatic area/total area = 0.5. I is the injected DC current and I _syn is the synaptic currents. I _L =g _L (V-E _L ). In our work, all of the ionic currents are modelled by the Hodgkin-Huxley type formalism, thus the dynamic of a gating variable x satisfies first-order kinetics,

$$ \dot{x} = {\phi_x}\left[ {{\alpha_x}(1 - x) - {\beta_x}x} \right] = {\phi_x}\left[ {{x_{\infty }} - x} \right]/{\tau_x} \dot{x} = {\phi_x}\left[ {{\alpha_x}(1 - x) - {\beta_x}x} \right] = {\phi_x}\left[ {{x_{\infty }} - x} \right]/{\tau_x} $$

(A3)

This equation will be used to calculate all of the gating variables.

Channel	Definition	Parameters
I Na	${g_{{Na}}}m_{\infty }^3h\left( {V - {E_{{Na}}}} \right) {g_{{Na}}}m_{\infty }^3h\left( {V - {E_{{Na}}}} \right)$	${m_{\infty }} = {\alpha_m}/\left( {{\alpha_m} + {\beta_m}} \right) {m_{\infty }} = {\alpha_m}/\left( {{\alpha_m} + {\beta_m}} \right)$
		${\alpha_m} = - 0.1\left( {V + 33} \right)/\exp \left[ { - 0.1\left( {V + 33} \right) - 1} \right] {\alpha_m} = - 0.1\left( {V + 33} \right)/\exp \left[ { - 0.1\left( {V + 33} \right) - 1} \right]$
		${\beta_m} = 4\exp \left[ { - \left( {V + 58} \right)/12} \right] {\beta_m} = 4\exp \left[ { - \left( {V + 58} \right)/12} \right]$
		${\alpha_h} = 0.07\exp \left[ { - \left( {V + 50} \right)/10} \right] {\alpha_h} = 0.07\exp \left[ { - \left( {V + 50} \right)/10} \right]$
		${\beta_h} = 1/\exp \left[ { - 0.1\left( {V + 20} \right) + 1} \right] {\beta_h} = 1/\exp \left[ { - 0.1\left( {V + 20} \right) + 1} \right]$
I K	${g_K}{n^4}\left( {V - {E_K}} \right) {g_K}{n^4}\left( {V - {E_K}} \right)$	${\alpha_n} = - 0.01(V + 34)/\exp [ - 0.1(V + 34) - 1] {\alpha_n} = - 0.01(V + 34)/\exp [ - 0.1(V + 34) - 1]$
		${\beta_n} = 0.125\exp \left[ { - \left( {V + 44} \right)/25} \right] {\beta_n} = 0.125\exp \left[ { - \left( {V + 44} \right)/25} \right]$
I Ca	${g_{{Ca}}}{m_{\infty }}\left( {V - {E_{{Ca}}}} \right) {g_{{Ca}}}{m_{\infty }}\left( {V - {E_{{Ca}}}} \right)$	${m_{\infty }} = 1/\exp \left[ { - \left( {V + 20} \right)/9} \right] {m_{\infty }} = 1/\exp \left[ { - \left( {V + 20} \right)/9} \right]$
I AHP	$\begin{gathered} {g_{{AHP}}}[C{a^{{2 + }}}]/([C{a^{{2 + }}}] + {K_D}) \hfill \\ (V - {E_K}) \hfill \\ \end{gathered} \begin{gathered} {g_{{AHP}}}[C{a^{{2 + }}}]/([C{a^{{2 + }}}] + {K_D}) \hfill \\ (V - {E_K}) \hfill \\ \end{gathered}$	$\frac{{d\left[ {C{a^{{2 + }}}} \right]}}{{dt}} = - \left[ {C{a^{{2 + }}}} \right]/{\tau_{{Ca}}} - B{I_{{Ca}}} \frac{{d\left[ {C{a^{{2 + }}}} \right]}}{{dt}} = - \left[ {C{a^{{2 + }}}} \right]/{\tau_{{Ca}}} - B{I_{{Ca}}}$
		${\tau_{{Ca}}} = 1000,B = 0.002,{K_D} = 30\mu M {\tau_{{Ca}}} = 1000,B = 0.002,{K_D} = 30\mu M$
I A	${g_A}{a^3}b(V - {E_K}) {g_A}{a^3}b(V - {E_K})$	${\alpha_a} = - 0.05(V + 20)/\{ \exp [ - (V + 20)/15] - 1\} {\alpha_a} = - 0.05(V + 20)/\{ \exp [ - (V + 20)/15] - 1\}$
		${\beta_a} = 0.1\left( {V + 10} \right)/\left\{ {\exp \left[ {\left( {V + 10} \right)/8} \right] - 1} \right\} {\beta_a} = 0.1\left( {V + 10} \right)/\left\{ {\exp \left[ {\left( {V + 10} \right)/8} \right] - 1} \right\}$
		${\alpha_b} = 0.00015/\exp \left[ {\left( {V + 18} \right)/15} \right] {\alpha_b} = 0.00015/\exp \left[ {\left( {V + 18} \right)/15} \right]$
		${\beta_b} = 0.06/\left\{ {\exp \left[ { - \left( {V + 73} \right)/12} \right] + 1} \right\} {\beta_b} = 0.06/\left\{ {\exp \left[ { - \left( {V + 73} \right)/12} \right] + 1} \right\}$
I CT	${g_{{CT}}}{c^2}d(V - {E_K}) {g_{{CT}}}{c^2}d(V - {E_K})$	$\begin{gathered} {\alpha_c} = - 0.0077(V + {V_{{shift}}} + 103)/ \hfill \\ \{ \exp [ - (V + {V_{{shift}}} + 103)/12] - 1\} \hfill \\ \end{gathered} \begin{gathered} {\alpha_c} = - 0.0077(V + {V_{{shift}}} + 103)/ \hfill \\ \{ \exp [ - (V + {V_{{shift}}} + 103)/12] - 1\} \hfill \\ \end{gathered}$
		${\beta_c} = 0.91 - {\alpha_c} {\beta_c} = 0.91 - {\alpha_c}$
		${\alpha_d} = 1/\exp \left[ {\left( {V + 79} \right)/10} \right] {\alpha_d} = 1/\exp \left[ {\left( {V + 79} \right)/10} \right]$
		${\beta_d} = 4/\left\{ {\exp \left[ { - \left( {V - 82} \right)/27} \right] + 1} \right\} {\beta_d} = 4/\left\{ {\exp \left[ { - \left( {V - 82} \right)/27} \right] + 1} \right\}$
		${V_{{shift}}} = 40\log \left( {\left[ {C{a^{{2 + }}}} \right]/13.805} \right) {V_{{shift}}} = 40\log \left( {\left[ {C{a^{{2 + }}}} \right]/13.805} \right)$
		${\tau_{{Ca}}} = 0.9,B = 0.06 {\tau_{{Ca}}} = 0.9,B = 0.06$

The values of the other parameters are ϕ = 4, g _L  = 0.1 and g _Ca  = 0.5 for soma and dendrite, g _Na  = 45, g _K  = 18, g _A  = 20 g _CT  = 140 and g _h  = 0.01 for soma and g _AHP  = 5, g _A  = 60 g _CT  = 70 and g _h  = 0.02 for dendrite; E _L  = −65, E _Na  = 55, E _K  = −80, E _Ca  = 120, and I _μ  = 4.9.

The OLM neuron is described as a single compartment model,

$$ \dot{V} = - {I_L} - {I_{{Na}}} - {I_K} - {I_{{Ca}}} - {I_h} - {I_{{AHP}}} - {I_{{syn}}} + I \dot{V} = - {I_L} - {I_{{Na}}} - {I_K} - {I_{{Ca}}} - {I_h} - {I_{{AHP}}} - {I_{{syn}}} + I $$

(A4)

Channel	Definition	Parameters
I Na	${g_{{Na}}}m_{\infty }^3h\left( {V - {E_{{Na}}}} \right)$	${\alpha_m} = - 0.1\left( {V + 35} \right)/\exp \left[ { - 0.1\left( {V + 35} \right) - 1} \right]$
		${\beta_m} = 4\exp \left[ { - \left( {V + 60} \right)/18} \right]$
		${\alpha_h} = 0.07\exp \left[ { - \left( {V + 58} \right)/20} \right]$
		${\beta_h} = 1/\exp \left[ { - 0.1\left( {V + 28} \right) + 1} \right]$
IK	${g_K}{n^4}\left( {V - {E_K}} \right)$	${\alpha_n} = - 0.01(V + 34)/\exp [ - 0.1(V + 34) - 1]$
		${\beta_n} = 0.125\exp \left[ { - \left( {V + 44} \right)/80} \right]$
ICa	${g_{{Ca}}}m_{\infty }^2\left( {V - {E_{{Ca}}}} \right)$	${m_{\infty }} = 1/\left\{ {\exp \left[ { - \left( {V + 20} \right)/9} \right] + 1} \right\}$
IAHP	${g_{{AHP}}}\left[ {C{a^{{2 + }}}} \right]/\left( {\left[ {C{a^{{2 + }}}} \right] + {K_D}} \right)\left( {V - {E_K}} \right)$	$\frac{{d\left[ {C{a^{{2 + }}}} \right]}}{{dt}} = - \left[ {C{a^{{2 + }}}} \right]/{\tau_{{Ca}}} - B{I_{{Ca}}}$
		${\tau_{{Ca}}} = 80,B = 0.002,{K_D} = 30\mu M$
Ih	${g_h}H\left( {V - {E_h}} \right)$	${H_{\infty }} = 1/\left\{ {\exp \left[ {\left( {V + 80} \right)/10} \right] + 1} \right\}$
		${\tau_H} = 200/\left\{ {\exp \left[ {\left( {V + 70} \right)/20} \right]} \right\} + \left. {\exp \left[ { - \left( {V + 70} \right)/20} \right] + 5} \right\}$

The other parameters are ϕ = 5, g _L  = 0.1, g _Na  = 35, g _K  = 9, g _AHP  = 10, g _Ca  = 1, gh = 0.15; E _L  = −65, E _Na  = 55, E _K  = −90, E _Ca  = 120, E _h  = −40, I _μ  = 0.

The basket neuron is described as a single compartment model,

$$ \dot{V} = - {I_L} - {I_{{Na}}} - {I_K} - {I_{{syn}}} + I \dot{V} = - {I_L} - {I_{{Na}}} - {I_K} - {I_{{syn}}} + I $$

(A6)

The parameters for calculation of the ionic currents are the same as that of OLM and I _μ  = 1.4.

The MSGABA neuron is described as a single compartment model,

$$ \dot{V} = - {I_L} - {I_{{Na}}} - {I_K} - {I_{{KS}}} - {I_{{syn}}} + I \dot{V} = - {I_L} - {I_{{Na}}} - {I_K} - {I_{{KS}}} - {I_{{syn}}} + I $$

(A7)

Channel	Definition	Parameters
INa	${g_{{Na}}}m_{\infty }^3h\left( {V - {E_{{Na}}}} \right)$	${\alpha_m} = - 0.1\left( {V + 33} \right)/\exp \left[ { - 0.1\left( {V + 33} \right) - 1} \right]$
		${\beta_m} = 4\exp \left[ { - \left( {V + 58} \right)/18} \right]$
		${\alpha_h} = 0.07\exp \left[ { - \left( {V + 51} \right)/10} \right]$
		${\beta_h} = 1/\exp \left[ { - 0.1\left( {V + 21} \right) + 1} \right]$
IK	${g_K}{n^4}\left( {V - {E_K}} \right)$	${\alpha_n} = - 0.01\left( {V + 38} \right)/\exp \left[ { - 0.1\left( {V + 38} \right) - 1} \right]$
		${\beta_n} = 0.125\exp \left[ { - \left( {V + 48} \right)/80} \right]$
IKS	${g_{{KS}}}pq\left( {V - {E_K}} \right)$	${p_{\infty }} = 1/\left\{ {\exp \left[ { - \left( {V + 34} \right)/6.5} \right] + 1} \right\}$
		${\tau_p} = 6$
		${q_{\infty }} = 1/\left\{ {\exp \left[ {\left( {V + 65} \right)/6.6} \right] + 1} \right\}$
		${\tau_q} = {\tau_{{q0}}}\left( {1 + 1/\left\{ {\exp \left[ { - \left( {V + 50} \right)/6.8} \right] + 1} \right\}} \right)$
		${\tau_{{q0}}} = 100$

The other parameters are ϕ = 5, g _L  = 0.1, g _Na  = 50, g _K  = 8, g _KS  = 12; E _L  = −50, E _Na  = 55, E _K  = −85, I _μ  = 22.

1.2 Synaptic connection definition

There are three types of synaptic neurotransmitters, the inhibitory GABA_A, the excitatory NMDA and AMPA. The GABA_A inhibitory post synaptic current (IPSC) is described as ${I_{{GAB{A_A}}}} = {g_{{syn}}}s\left( {V - {E_{{GAB{A_A}}}}} \right) {I_{{GAB{A_A}}}} = {g_{{syn}}}s\left( {V - {E_{{GAB{A_A}}}}} \right)$, where the activation variable s is calculated by $\dot{s} = \alpha F\left( {{V_{{pre}}}} \right)\left( {1 - s} \right) - \beta s \dot{s} = \alpha F\left( {{V_{{pre}}}} \right)\left( {1 - s} \right) - \beta s$. The V _pre is the presynaptic neuron membrane potential, $F\left( {{V_{{pre}}}} \right) = 1/\left[ {1 + \exp \left( { - {V_{{pre}}}/K} \right)} \right] F\left( {{V_{{pre}}}} \right) = 1/\left[ {1 + \exp \left( { - {V_{{pre}}}/K} \right)} \right]$. Parameters for different neurons couples are:

basket-pyramidal (b-p)	α = 10, β = 0.1, K = 2, ${E_{{GAB{A_A}}}} = - 80$, g sym  = 2.76
OLM-basket (o-b)	α = 20, β = 0.1, K = 2, ${E_{{GAB{A_A}}}} = - 80$, g sym  = 1.76
OLM-pyramidal (o-p)	α = 20, β = 0.1, K = 2, ${E_{{GAB{A_A}}}} = - 85$, g sym  = 1.76
OLM-MSGABA (o-m)	α = 20, β = 0.1, K = 0.5, ${E_{{GAB{A_A}}}} = - 80$, g sym  = 0.5
basket-basket (b-b)	α = 10, β = 0.1, K = 2, ${E_{{GAB{A_A}}}} = - 75$, g sym  = 0.125
MSGABA-OLM (m-o)	α = 10, β = 0.1, K = 2, ${E_{{GAB{A_A}}}} = - 75$, g sym  = 0.5
MSGABA-MSGABA (m-m)	α = 10, β = 0.1, K = 2, ${E_{{GAB{A_A}}}} = - 75$, g sym  = 0.25
MSGABA-basket (m-b)	α = 10, β = 0.1, K = 2, ${E_{{GAB{A_A}}}} = - 75$, g sym  = 1

The AMPA and NMDA excitatory post synaptic current (EPSP) are described as ${I_{{AMPA}}} = {g_{{AMPA}}}s\left( {V - {E_{{AMPA}}}} \right) {I_{{AMPA}}} = {g_{{AMPA}}}s\left( {V - {E_{{AMPA}}}} \right)$ and ${I_{{NMDA}}} = {g_{{NMDA}}}B(V)s\left( {V - {E_{{NMDA}}}} \right) {I_{{NMDA}}} = {g_{{NMDA}}}B(V)s\left( {V - {E_{{NMDA}}}} \right)$, respectively. s is updated as $\dot{s} = \alpha [T](1 - s) - \beta s \dot{s} = \alpha [T](1 - s) - \beta s$, $[T] = {T_{{\max }}}/\left[ {1 + \exp \left( { - {V_{{pre}}} + {V_p}} \right)/{K_p}} \right] [T] = {T_{{\max }}}/\left[ {1 + \exp \left( { - {V_{{pre}}} + {V_p}} \right)/{K_p}} \right]$, V _p  = 2, V _K  = 5, B(V) is calculated by $B(V) = 1/\left\{ {1 + \exp \left( { - 0.062V} \right)\left[ {M{g^{{2 + }}}} \right]/3.5} \right\} B(V) = 1/\left\{ {1 + \exp \left( { - 0.062V} \right)\left[ {M{g^{{2 + }}}} \right]/3.5} \right\}$, where [Mg ²⁺ ] = 1 mM by default. The α and β for AMPA. and NMDA are α = 1.1, β = 0.19 and α = 0.072, β = 0.0066, respectively. ${E_{{AMPA}}} = {E_{{NMDA}}} = 0 {E_{{AMPA}}} = {E_{{NMDA}}} = 0$. From pyramidal to basket neurons g _AMPA  = 0.1, and from pyramidal to OLM neurons g _AMPA  = 1.35 and g _AMPA  = 0.625. The summated synaptic current is normalized by the number of presynaptic neurons.