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

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DOI: 10.1007/s10827-011-0363-7

- Cite this article as:
- Zou, X., Coyle, D., Wong-Lin, K. et al. J Comput Neurosci (2012) 32: 465. doi:10.1007/s10827-011-0363-7

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## 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.

### Keyword

Alzheimer’s diseaseβ-AmyloidA-type K^{+}currentPhase-plane analysisHippocampo-septal theta rhythm

## 1 Introduction

Alzheimer’s disease (AD) is a neurodegenerative disease associated with memory deficits and cognitive decline, which are caused by pathological changes in the brain (Minati et al. 2009). In the early stage of AD, the hippocampus in the medial temporal lobe is one of the first regions to be affected, especially the hippocampal pyramidal neurons (Adeli et al. 2005; Li et al. 2010). Damage to the hippocampus can lead to profound amnesia. Hippocampo-septal theta rhythm (4–7 Hz) has been found to play a critical role in memory processing, and theta rhythm abnormalities is usually related to memory deficits and pathological changes in brain (Vertes 2005; Colom 2006). Therefore, we will evaluate the hippocampus dynamical abnormalities in terms of the hippocampo-septal theta band power changes.

AD is characterized by two neuropathological structures: neurofibrillary tangles and senile plaques (Tiraboschi et al. 2004). Neurofibrillary tangles are caused by the microtubule-binding protein, tau, becoming hyperphosphorylated. Senile plaques are mainly composed of beta-amyloid (Aβ). Aβ acts as a neurotoxin causing neuronal dysfunction and apoptosis (Hardy and Higgins 1992) and synaptic plasticity impairment (Holscher et al. 2007). Aβ may change the activity of various ionic channels in pyramidal neurons, e.g., L-type Ca^{2+} channels (I_{Ca}) (Webster et al. 2006); A-type fast-inactivating K^{+} channels (I_{A}) and delayed rectifying K^{+} channels (I_{K}) (Good et al. 1996); large-conductance Ca^{2+}-activated K^{+} channels (I_{CT}) (Ye et al. 2010; Chi and Qi 2006). In addition, Aβ can disturb the neuromodulators, (Tran et al. 2002; Palop and Mucke 2010), which control memory encoding and recall (Hasselmo et al. 1996). As Aβ precedes tau protein in the progression of AD (Takahashi et al. 2010), this work focuses on the study of the impact of Aβ on the hippocampus.

The aim of this paper is to investigate the impacts of Aβ-induced pathological changes on hippocampal neural dynamics through computational studies. Extensive computational work has been performed to investigate the AD-induced hippocampal memory encoding, retrieval difficulties and pathological changes in synapses (Adeli et al. 2005). Building on previous work, we have constructed an integrated computational model of the hippocampal CA1 region coupled to the medial septum, which includes pyramidal, basket, OLM and MSGABA neurons. We identify the significance of the impact of Aβ on the hippocampal network dynamics by evaluating changes in the hippocampo-septal theta band power. In our previous work, we have evaluated various ionic channels in the hippocampal pyramidal neurons, which have been reported to be affected by Aβ. Specifically, changes in I_{Ca}, I_{A}, I_{K} and I_{CT} have been evaluated and we found that only Aβ-blocked I_{A} has induced a significant theta band power change (Zou et al. 2011), probably due to the increased pyramidal neuronal excitability (Morse et al. 2010). In this paper, a wider range of model parameters of I_{A} will be explored and the corresponding theta band power changes will be evaluated. Then the mechanisms underlying the Aβ-blocked I_{A} induced pyramidal neuronal excitability increases and its relationship to the theta band power changes will be mathematically and systematically investigated. We will also explore possible implications of our results to memory disorder and epilepsy.

## 2 Method

### 2.1 Conductance-based neural network

*in vivo*experiments (Csicsvari et al. 1999; Ylinen et al. 1995; Klausberger et al. 2003) and simulations (Wang 2002; Rotstein et al. 2005). The schematic illustration of the network architecture is shown in Fig. 1. Each type of neuron in Fig. 1 represents a population of identical neurons. The network theta oscillations are generated from the intrinsic theta oscillations of MSGABA neurons (Stewart and Fox 1990; Cobb et al. 1995; Wang 2002; Freund and Antal 1988; Toth et al. 1997).

_{Na}and I

_{K}and the dendrite contains a calcium dependent potassium current I

_{AHP}. Both the soma and the dendrite contain leakage currents I

_{L}and high-threshold L-type calcium currents I

_{Ca}. Furthermore, the pyramidal neurons in the hippocampus CA1 contain other types of ionic currents to account for neuronal functions (Warman et al. 1994). In this work, we will incorporate some of those currents which have been shown to be affected by Aβ. As a result, our model also contains A-type transient potassium currents I

_{A}and large-conductance calcium dependent potassium currents I

_{CT}in the soma and dendrite, respectively. The membrane potential dynamics are modelled as follows:

*V*denotes membrane potential, subscript

*s*and

*d*denote soma and dendrite, respectively. The synaptic current

*I*

_{syn}represents the synaptic interactions from other neurons in the network. g

_{c}is the coupling conductance between soma and dendrite, and

*p*=soma area/total area, with 0.5 as the default value. To emulate heterogeneity in the brain, the injected DC current,

*I*, for each neuron is not chosen to be identical. This is achieved by allowing

*I*to follow a Gaussian distribution with mean

*I*

_{μ}and standard derivation

*I*

_{σ}. For the pyramidal neuronal population,

*I*

_{μ}= 4.9

*μA*/

*cm*

^{2}and

*I*

_{σ}= 0.1

*μA*/

*cm*

^{2}.

*I*

_{Na},

*I*

_{K},

*I*

_{L},

*I*

_{Ca}, hyperpolarization activated current

*I*

_{h}and

*I*

_{AHP}Eq. (3). The model of a basket neuron has

*I*

_{Na},

*I*

_{K}, and

*I*

_{L}Eq. (4). The MSGABA neuronal model contains

*I*

_{Na},

*I*

_{K},

*I*

_{L}and a slowly inactivating potassium current

*I*

_{KS}Eq. (5).

For the basket, OLM and MSGABA neuronal populations, \( {I_{\mu }} = 1.4\mu A/c{m^2},\,0\mu A/c{m^2},\,2.2\mu A/c{m^2} {I_{\mu }} = 1.4\mu A/c{m^2},\,0\mu A/c{m^2},\,2.2\mu A/c{m^2} \), respectively; \( {I_{\sigma }} = 0.1\mu A/c{m^2} {I_{\sigma }} = 0.1\mu A/c{m^2} \) for all populations. Details of the definition of all the other parameters are provided in the Appendix.

We found from our simulations that the changes in the connection topology do not significantly affect the dynamics of the network and our results, as far as the number of synapses per cell is larger than a critical value (Wang and Buzsaki 1996). Therefore, the simulation results presented in this paper are based on an all-to-all connectivity, because the exclusion of heterogeneity in the connectivity makes our analysis more tractable. The pyramidal neurons innervate basket neurons via AMPA-type receptor and OLM via AMPA- and NMDA-type receptors, the other synaptic connections are mediated by the GABA_{A}-type receptors. The number of pyramidal, basket, OLM and MSGABA neurons is 10, 100, 30 and 50, respectively. The ratio of interneurons is based on that reported in (Freund and Buzsaki 1996). Although the number of pyramidal neurons is large in the hippocampus, a limited number of pyramidal neurons were used in (Hajós et al. 2004) for the purpose of computational efficiency. It has been shown that such limited numbers of pyramidal neurons are sufficient to yield acceptable statistics and thus this setting is used in our work. Noise in the membrane potential is generated by following a Gaussian distribution with zero mean and standard derivation of 1.1 *μA*/*cm*^{2}. The membrane noise is randomly generated in each trial. Note that membrane noise is excluded in Section 3.2 for analysis.

*p*< 0.05.

### 2.2 A reduced pyramidal neuron model for neuronal dynamical analysis

A useful way to analyze a dynamical (neural) system is to construct its phase plane (also known as phase portrait) (Izhikevich 2007). A phase plane can depict the stability of the system such that its dynamics can be rigorously analyzed and understood geometrically. However, the limitation of the phase-plane analysis is that it is difficult to be applied for a system that contains multiple dynamical variables. The complexity of the two-compartmental model of pyramidal neurons in the network makes it unsuitable to be analyzed using phase-plane analysis, because it contains too many dynamical variables.

*I*

_{Na},

*I*

_{K}and

*I*

_{L}from the soma and

*I*

_{A}from the dendrite in the full model. The dynamical variables in the reduced one-compartmental pyramidal model can be further reduced by performing additional approximations. First, as the activation gating process of

*I*

_{Na}and

*I*

_{A}, denoted by

*m*and

*a*, respectively, are very fast, they can be considered as being relatively instantaneous, therefore their steady-state value

*m*

_{∞}and

*a*

_{∞}can be used. Second, there is an approximately linear relationship between the inactivation gating variable of

*I*

_{Na},

*h*, and the activation gating variable of

*I*

_{K},

*n*, i.e.,

*h = 0.89–1.1n*(Izhikevich 2007). Based on the above approximations, the final reduced dynamics of the pyramidal neuron becomes:

Note that Eq. (6) only consists of three dynamical variables, i.e., *V*, *n* and *b*. The definition of the parameters is the same as they are defined in Eqs. (1 and 2) (see the Appendix for further details).

The aim of the model reduction approximation is to reduce the number of dynamical variables but not to significantly change the system statistical characteristics. To verify that the reduced model dynamics do not deviate significantly from the full model, results from 15 trials running with original parameter settings were obtained for both the full network and the network employing the reduced pyramidal neuron model implementations. To compensate for the changes in the pyramidal neurons model, *I*_{μ} are modified as 3.5 *μA*/*cm*^{2}, −0.4 *μA*/*cm*^{2}, 1 *μA*/*cm*^{2} and 2.2 *μA*/*cm*^{2} for pyramidal, OLM, basket and MSGABA populations, respectively. A one-way ANOVA test was performed and the results verify that the model reduction process does not significantly change theta band power (*p* = 0. 6557). Note that the simplified pyramidal neuronal model is applied in the phase-plane analysis presented in Section 3.2.1.

## 3 Results

### 3.1 Network simulations

*T*_{1}: Early in time epoch T_{1}, the pyramidal neurons start to spike which evoke the OLM and basket neurons quickly via fast AMPA-type receptors. The subsequent spikes of the OLM neurons are promoted by the NMDA-type receptors, which in turn suppress the pyramidal and basket neurons. After the OLM neurons have spiked, the basket and pyramidal neurons are gradually depolarized by the injected depolarization currents.*T*_{2}: As the basket neurons depolarize faster than the pyramidal neurons, they will spike earlier and inhibit the pyramidal neurons via the GABA_{A}-type receptors. During the course of time epoch T_{2}, the basket neurons keep spiking and the pyramidal neurons are hyperpolarized.*T*_{3}: At the beginning of epoch T_{3}, the MSGABA neurons start to spike, which inhibit basket neurons and release the pyramidal neurons. The spikes of the MSGABA neurons are stopped by the firing of OLM neurons that have been evoked by the pyramidal neurons.

Such a spiking phase relationship is consistent with previous experimental and computational studies. In (Klausberger et al. 2003; Cutsuridis et al. 2010; Klausberger and Somogyi 2008), it has been shown that the pyramidal and OLM neurons spike in phase, while the basket neurons spike out of phase with them. Wang (2002) has shown that the OLM and MSGABA neurons spike out of phase in theta oscillation.

_{A}in dendrite is decreased to simulate the Aβ-blocked I

_{A}in the dendrite (Chen 2005; Morse et al. 2010; Good et al. 1996; Xu et al. 1998; Zhang and Yang 2006). The obtained theta band power with different values of g

_{A}is illustrated in Fig. 4. Interestingly, it can be seen that with the decrease in g

_{A}, the theta band power first slightly increases to a maximum value with 0.9 g

_{A}before it starts to decrease. When g

_{A}is below 0.3 g

_{A}, the theta band power almost disappears. Figure 4 demonstrates that decreasing g

_{A}initially causes an increase in theta oscillations, after which a further decrease in g

_{A}produces a corresponding decrease in the theta activity. To understand the network’s dynamics underlying such observations, the dynamics of a single reduced pyramidal neuron and the whole network are systematically analyzed in Section 3.2.

### 3.2 Mechanisms underlying the changes in the network dynamics

The simulation results presented in the previous section show that decreased g_{A} induces theta band power changes. In our previous work (Zou et al. 2011), we have shown that the increased theta band power is due to the increased synchrony of the pyramidal neuronal population. The network theta oscillation is originated from MSGABA neurons, and the theta oscillation of pyramidal neurons is driven by the inhibitory post synaptic currents from the interneurons. During the trough of a pyramidal theta cycle, the inhibition on the pyramidal neurons is large; therefore the pyramidal neurons are mostly quiescent. But during the peak or crest of the pyramidal theta cycle, as inhibition from the interneurons decreases, the pyramidal neurons will be able to spike; and the neuronal dynamics (to spike or not) is dependent on the received depolarization currents and the neuronal excitability. As can be inferred from Fig. 3 (epoch T3) that the inhibition afference is low and relatively constant during this time epoch, the pyramidal neuronal dynamics are mainly dependent on the intrinsic neuronal excitability. The higher the neuronal excitability is, the easier the pyramidal neurons can spike (resulting in higher population synchrony). Therefore, we are justified to use a reduced single pyramidal neuron model to analyze the pyramidal neuronal excitability. This will be performed in Section 3.2.1. Furthermore, the relationship between the hyper-excitability of pyramidal neurons and the decreased theta band power will be investigated in Section 3.2.2.

#### 3.2.1 Aβ-blocked I_{A} increases neuronal excitability

To investigate the enhanced neuronal excitability as a result of decreased g_{A}, a dynamical systems analysis of a single reduced pyramidal neuron modelled by Eq. (6) was performed. The applied DC current, *I*, in Eq. (6) is reduced to 2 *μA*/*cm*^{2} to compensate for the absence of inhibition from interneurons. As mentioned in Section 2.2, the reduced pyramidal neuron model contains three dynamical variables *V*, *n* and *b*. Therefore, the time evolution of its state can be plotted as trajectories in a three-dimensional (phase) space—a stereoscope image (Izhikevich 2007).

_{A}as an example in this section instead of using 0.9 g

_{A}. Figure 5 plots the (noiseless) trajectories in the original and 0.8 g

_{A}conditions, with either a high or low initial value of its membrane potential

*V*

_{0}. In Fig. 5, the initial values of

*b*and

*n*are all zero. Figure 5 (a and b) are obtained in the original g

_{A}condition; c and d are obtained with the value of g

_{A}set at 0.8 times its original value. From the plots, it can be clearly seen that in the original g

_{A}condition, the tendency to spike (an increase followed by a decrease in

*V*) is highly dependent on its initial state. With a low initial membrane potential value of

*V*

_{0}= −65 mV, the neuron will quickly converge to a stable state without spiking (Fig. 5(a)). If the initial membrane potential value is higher (

*V*

_{0}= −45 mV), the neuron produces a spike and then converges back to the stable state (Fig. 5(b)). In the case of 0.8 g

_{A}, the dynamics of the neuron become less dependent on its initial state. In Fig. 5 (c and d), the neuron is capable of spiking with either a high or low initial membrane potential. Note that it also spikes repetitively.

Under the original g_{A} condition and in the presence of noise, the probability of a pyramidal neuron spiking is reduced by the presence of the stable state. However in the case of low g_{A}, pyramidal neurons become more susceptible to spiking as there is no such stable state in the system. This explains the increased excitability of pyramidal neurons with low g_{A}. Although Fig. 5 shows some principle dynamical characteristics of the pyramidal neurons associated with a decrease in g_{A}, it is more rigorous to visualize and analyze the above mechanisms in a two-dimensional phase space. For example, the transition from resting (Fig. 5(a)) to repetitive spiking (Fig. 5(c)) could be induced by a bifurcation (abrupt qualitative change in behaviour). To clearly demonstrate the properties of such a bifurcation, a phase-plane analysis is employed by producing two-dimensional slices of the stereoscope images from Fig. 5.

*b*is an order of magnitude slower than

*n*at low membrane potential and almost unchanged at high membrane potential. Therefore a two-dimensional slice of the stereoscope image can be approximated by fixing

*b*at a specific value, and plotting the dynamical trajectories in the (

*V*,

*n*) space. Nullclines in the (

*V*,

*n*) space, i.e.,

*V*- and

*n*-nullclines, are obtained by separately solving two equations, \( \mathop{V}\limits^{ \cdot } = 0 \mathop{V}\limits^{ \cdot } = 0 \) and \( \mathop{n}\limits^{ \cdot } = 0 \mathop{n}\limits^{ \cdot } = 0 \), such that

*m*

_{∞},

*a*

_{∞},

*n*

_{∞}and

*τ*

_{∞}are all functions of

*V*, as they are defined in the Appendix.

*g*

_{L},

*g*

_{Na},

*g*

_{K},

*g*

_{A},

*E*

_{L},

*E*

_{Na},

*E*

_{K}and

*ϕ*

_{n}are all constants.

*b*

_{c}denotes the variable

*b*fixed as a constant. By definition, intersections of the nullclines provide the equilibria, i.e., Steady States of the system. The stability of an equilibrium point (

*V*

_{ss},

*n*

_{ss}) can be evaluated by analysing the Jacobian matrix of the system \( \dot{V} \dot{V} \) and \( \dot{n} \dot{n} \) at that point (Izhikevich 2007).

*b*is no larger than 0.06. Therefore, it is sufficient to evaluate

*b*

_{c}

*< 0.06*. The obtained phase spaces under different conditions are illustrated in Fig. 6. It can be seen from Fig. 6(a) that when

*b*

_{c}

*= 0*, the

*V*- and

*n*-nullclines (denoted by solid and dished lines, respectively) have only one intersection, the system is unstable and a stable limit cycle attractor (corresponding to repetitive spiking) exists. In such case, the system will converge onto the limit cycle from any initial state. As a result, the membrane potential will keep changing along the limit cycle without termination (Fig. 6(a), bold line). However, it has been shown in Fig. 5 that with low membrane potentials, the gating variable

*b*can quickly increase from zero. Such increase shifts the lower deflection point of the

*V*-nullcline downwards but does not change the

*n*-nullcline. As a result, the

*V*-nullcline additionally intersects with the

*n*-nullcline at the low-voltage range and forms a new equilibrium at

*b*

_{c}= 0.027 (Fig. 6(b)). At this critical point, the equilibrium is actually a coalescence of a stable node (corresponding to a stable steady-state) and an unstable equilibrium (saddle). With further increase in

*b*, the node and saddle split, and the system now has a stable steady state (Fig. 6(c)). In this case, the neuron cannot produce repetitive spiking as trajectories of neuronal dynamics always terminate at the stable steady-state, as shown in Fig. 6(c). This procedure is referred to as a saddle-node bifurcation, which results in a transition of the system from periodic spiking to resting dynamics. This explains the trajectories illustrated in Fig. 5(a and b). With low initial membrane potential, the gating variable

*b*quickly activates and the bifurcation occurs, which produces a stable node. The dynamical trajectory is attracted by the stable node without spiking (Fig. 5(a)). However, with a high initial membrane potential, the gate

*b*is kept inactive at the beginning. At this moment, the system only has a stable limit cycle. The system’s trajectory is attracted by this limit cycle and produces a spike. At the end of the spike, the gate

*b*activates which leads to a bifurcation. The trajectory is then attracted to the stable node (Fig. 5(b)). When g

_{A}is sufficiently low, the

*V*-nullcline is prevented from shifting downwards due to the increase in

*b*. As a result, the saddle-node bifurcation may not happen with the increase in

*b*(Fig. 6(d, e and f)). In such cases, the system only has a limit cycle for various values of

*b*, which explains the trajectories demonstrated in Fig. 5(c and d).

#### 3.2.2 Hyper neuronal excitability can cause the disappearance of theta rhythm

_{A}, the pyramidal and OLM neurons spike in phase, while MSGABA and basket neurons do not spike at all. Therefore, measuring the oscillation frequency of pyramidal neurons is sufficient to provide an approximation of the network oscillation. Figure 7(a) shows the oscillation frequency of the pyramidal neurons in the noise-free condition. It can be seen that there exists a critical point corresponding to the value of g

_{A}between 0.3–0.4 times its normal value. At larger values of g

_{A}, the pyramidal neurons oscillate at the theta rhythm (Fig. 7(a) left and middle insets). But at smaller values of g

_{A}, theta oscillations abruptly disappear with the pyramidal neurons spiking at a high frequency (Fig. 7(a) right inset), which gradually increases with decreasing g

_{A}. The slight frequency drop with 0.5 and 0.4 g

_{A}is due to the pyramidal neurons producing multiple spikes around the peak of the theta cycle caused by high excitability (Fig. 7(a) middle inset), which expand the oscillation period and decrease the frequency. At this time, theta frequency and high frequency co-exist in the network oscillation. Figure 7(b) shows an example of the transition of the dominant frequency with g

_{A}decrease.

_{A}decreasing, the lower deflection point of the

*V*-nullcline is prevented from shifting towards the

*n*-nullcline. The lower g

_{A}is, the greater the distance from the

*V*-nullcline to the

*n*-nullcline will be. As a result, the depolarization of the pyramidal neurons becomes faster (Guckenheimer and Holmes 1997; Izhikevich 2007). The decreased g

_{A}induced faster depolarization is also illustrated in Fig. 8(b). When g

_{A}is as low as 0.3 times its original value, the pyramidal neurons will always depolarize faster than the basket neurons, which results in relatively earlier spiking of pyramidal neurons compared to that of basket neurons, thus swopping their phase relationship within a theta cycle. This will evoke the spikes of the OLM neurons via AMPA and NMDA, which prevents the basket neurons from spiking. Without inhibition from the basket neurons, the pyramidal neurons will depolarize quickly again and produce more spikes. As a result, the pyramidal neurons will consecutively spike at a higher frequency and the network oscillation will only be dependent on the spiking of the pyramidal neurons, but not on the spiking phase relationships among the different neuronal populations. As the depolarization speed of the pyramidal neurons keeps increasing with decreasing g

_{A}, the oscillation frequency of pyramidal neurons increases slightly.

To verify our hypothesis on the causal relationship between changes in the pyramidal neurons and basket neurons, we artificially injected brief depolarization current pulses into the pyramidal neurons just before the spikes of the basket neurons in the noise-free original g_{A} condition. Given our hypothesis, we would expect similar results to those obtained under the very low g_{A} condition. A brief current pulse with a duration of 3 ms and amplitude of 50 *μA*/*cm*^{2} was applied every 55 ms. 200 ms simulation results selected from single-trials are presented in Fig. 8(c). As anticipated, membrane potential dynamics observed in the original g_{A} condition were very similar to those observed in the very low g_{A} condition.

## 4 Discussion

In our previous work (Zou et al. 2011), we have investigated the impact of Aβ-induced changes in various ionic channels of hippocampal pyramidal neurons on system dynamics by evaluating theta band power changes. We have found that only Aβ blocked I_{A} can significantly alter theta band power. In this paper, we explored a wider range of the model parameters of I_{A} and found Aβ blocked I_{A} induced both increase and decrease in theta band power. Both the theta band power changes were induced by the enhanced neuronal excitability. Analysis based on a reduced pyramidal neuronal model showed that the decreased I_{A} prevented the emergence of a steady state of the neuron and in turn increased the neuronal excitability. As we have shown in Fig. 3, the network theta oscillation consisted of various neuronal populations spiking at different phases of a theta cycle. Supposing a theta cycle starts from the spiking of pyramidal neurons, the spiking phase of different neuronal populations within the theta cycle should be in the temporal order: the pyramidal and OLM neurons, followed by the basket neurons, and finally the MSGABA neurons. With moderate low g_{A}, such spiking relationship was kept; only more pyramidal neurons produced spikes during the peak of pyramidal theta oscillation, which increased the synchrony of the pyramidal population and thus increased the theta band power. However, when the pyramidal neurons became hyper-excited, the spiking phase relationship was violated and the theta band power decreased. Interestingly, the decreased theta band power is consistent with an experimental finding on animal model of AD (Mugantseva and Podolski 2009).

_{A}induced memory deficits is illustrated in Fig. 9. Suppose a single item (1 0 1 1 1 0 1 0 1 1) in the memory is represented by neuronal dynamics over all the 10 pyramidal neurons, by depolarizing the selective neurons. Then, under the control condition, the memory can be correctly presented over time by the periodic spiking patterns of the pyramidal neurons (Fig. 9(a)). With lower g

_{A}, undesired neurons are also incorporated into spiking, possibly resulting in either a wrong memory or multiple interfering memories being represented (Fig. 9(b)). With very low g

_{A}, all pyramidal neurons become hyper-excited and the theta rhythm ceases to exist (Fig. 9(c)). Indeed, a recent study has shown that Aβ could be the main cause of epilepsy in AD due to hippocampal network hyper-excitability (Palop et al. 2007). Our work supports such an observation.

With regard to the mechanisms underlying theta rhythm generation in the hippocampus, there exist multiple hypotheses. One theory hypothesized that the Medial Septem GABAergic afference imposes the theta oscillation on the GABAergic cells in the hippocampus and in turn on the firing of pyramidal neurons. This hypothesis has been supported by experimental and theoretical studies (Stewart and Fox 1990; Cobb et al. 1995; Wang 2002; Freund and Antal 1988; Toth et al. 1997). However, some other studies showed that the slow deactivation current I_{h} in the pyramidal neurons and OLM neurons may also contribute to theta oscillation by providing rebound action potentials (Rotstein et al. 2005; Orban et al. 2006). Orban et al. (2006) have demonstrated that I_{h} in the pyramidal neurons act as a pacemaker of the theta rhythm in a hippocampus standalone network. In our model, we followed the former hypothesis, i.e., hippocampo-septal theta oscillation is originated from MSGABA neurons. Furthermore, there is no experimental evidence showing that I_{h} in the hippocampus is affected by Aβ. Therefore we have excluded the I_{h} in the pyramidal neurons. Nevertheless, the comparison of the effects of these two hypotheses will be an interesting project on its own, but it is beyond the scope of this paper. In our work, the simulations and analyses were based on an all-to-all network connectivity, as heterogeneity in the neuronal connections may cause some difficulties in our analysis. For example, it will be more difficult to sort, evaluate and compare the depolarization time of various neurons. Nevertheless, as shown in (Wang and Buzsaki 1996; Golomb and Hansel 2000), as long as the number of synapses per cell is larger than a critical value, the neuronal network dynamics remain robust.

To conclude, we have shown that Aβ blockage of I_{A} could be an important reason for causing behavioural changes in the hippocampal network. Although in this work, we only focus on the system dynamical changes caused by the pathological changes in the hippocampal pyramidal neurons, it should be admitted that the Aβ-indued hippocampal dysfunction may be more complex. For example, it has been shown in (Villette et al. 2010; Colom et al. 2010) that Aβ-induced GABAergic and cholinergic system degeneration in the medial septum may also cause hippocampal dysfunctions. GABAergic neurons inhibit hippocampal OLM and basket neurons and act as a hippocampal theta rhythm pacemaker and cholinergic neurons control the acetylcholine level in the hippocampus, which can change the dynamics of pyramidal neurons and synapses. Investigating these factors will be a topic of future research.

## Acknowledgement

This study is currently supported under the CNRT award by the Northern Ireland Department for Employment and Learning through its “Strengthening the All-Island Research Base” initiative. We are grateful to Dr. Christian Hölscher for comments on an earlier version of our manuscript.